A LevelMathsEdexcelPureExam QuestionsVectorsVectors in 3D

Vectors in 3D (Edexcel A Level Maths: Pure): Exam Questions

Exam code: 9MA0

4 hours41 questions
Easy
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Hard
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1
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3 marks

Relative to a fixed origin O, the points A and B have coordinates \left(3, -4, 2\right) and \left(-5, 2, -8\right) respectively.

Find the exact distance between A and B, giving your answer in the form a\sqrt{b}, where a and b are integers to be found.

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2a
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2 marks

Figure 1 shows a sketch of triangle ABC.

scAfNotO_q9-11-1-vectors-in-2-dimensions-easy-a-level-maths-pure
Figure 1

Given that

\overrightarrow{AB} = \mathbf{i} + 4\mathbf{j} - 2\mathbf{k}

\overrightarrow{AC} = 6\mathbf{i} - 2\mathbf{j} + 8\mathbf{k}

Find \overrightarrow{BC}.

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2b
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2 marks

Hence, or otherwise, find the distance BC, giving your answer to three significant figures.

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3a
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3 marks

Relative to a fixed origin O, the point P has coordinates \left(2, -1, -5\right) and the point Q has coordinates \left(-6, -12, 11\right).

Find the vector \overrightarrow{PQ} and hence find the exact distance PQ.

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3b
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2 marks

Find a unit vector in the direction of \overrightarrow{PQ}, giving your answer in its simplest form.

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4a
1 mark

Relative to a fixed origin O, the point A has coordinates \left(-3, 4, 9\right).

The vectors \overrightarrow{AB} and \overrightarrow{CD} are given by

\overrightarrow{AB} = \begin{pmatrix} 4 \\ -5 \\ 2 \end{pmatrix} \quad \text{and} \quad \overrightarrow{CD} = \begin{pmatrix} -2 \\ 2.5 \\ -1 \end{pmatrix}

Find the coordinates of the point B.

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4b
2 marks

State, giving a reason, whether the vectors \overrightarrow{AB} and \overrightarrow{CD} are parallel.

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5
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4 marks

The points P, Q and R have coordinates \left(4, -3, 12\right), \left(3, -7, 9\right) and \left(7, -9, 15\right) respectively.

Determine whether triangle PQR is scalene, isosceles or equilateral. Fully justify your answer.

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6
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4 marks

The vectors \mathbf{p} and \mathbf{q} are defined by

\mathbf{p} = 14\mathbf{i} + (a + b)\mathbf{j} + (c - b + 1)\mathbf{k}

\mathbf{q} = a\mathbf{i} + 6\mathbf{j} - 4\mathbf{k}

where a, b and c are scalar constants.

Given that \mathbf{p} = 2\mathbf{q}, find the value of a, the value of b and the value of c.

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7a
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3 marks

A particle P of mass 0.5\text{ kg} is acted upon by a force \mathbf{F} where

\mathbf{F} = \left(3\mathbf{i} - 5\mathbf{j} - 2\mathbf{k}\right)\text{ N}

(i) Find the acceleration of P.

(ii) Hence find the magnitude of the acceleration of P, giving your answer to 3 significant figures.

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7b
2 marks

A second force \mathbf{G} now acts on P. Given that the resultant of \mathbf{F} and \mathbf{G} is \left(4\mathbf{i} + 2\mathbf{k}\right)\text{ N}, find \mathbf{G} in the form x\mathbf{i} + y\mathbf{j} + z\mathbf{k}, where x, y and z are constants to be found.

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8a
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3 marks

Two forces \mathbf{F}_1 and \mathbf{F}_2 act on a particle of mass 10\text{ kg}. The forces are given by

\mathbf{F}_1 = \left(2\mathbf{i} + p\mathbf{j} - 8\mathbf{k}\right)\text{ N}

\mathbf{F}_2 = \left(q\mathbf{i} + 3q\mathbf{j} + (p - q)\mathbf{k}\right)\text{ N}

where p and q are scalar constants.

Under the action of these two forces, the particle is in equilibrium.

(i) Find the value of p and the value of q.

(ii) Explain how you can verify your answer to part (a)(i).

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8b
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5 marks

A third force \mathbf{F}_3 = \left(p\mathbf{i} + q\mathbf{j} + pq\mathbf{k}\right)\text{ N} is now applied to the particle.

Find:

(i) the resultant force \mathbf{R} now acting on the particle,

(ii) the acceleration of the particle,

(iii) the magnitude of the acceleration of the particle, giving your answer to 3 significant figures.

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1a
2 marks

Relative to a fixed origin O

  • point A has position vector 2 bold i plus 5 bold j minus 6 bold k

  • point B has position vector 3 bold i minus 3 bold j minus 4 bold k

  • point C has position vector 2 bold i minus 16 bold j plus 4 bold k

Find stack A B with rightwards arrow on top.

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1b
2 marks

Show that quadrilateral O A B C is a trapezium, giving reasons for your answer.

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2a
1 mark

Relative to a fixed origin O

  • the point A has position vector 5 bold i plus 3 bold j plus 2 bold k

  • the point B spacehas position vector 2 bold i plus 4 bold j plus a bold k

where a is a positive integer.

Show that open vertical bar stack O A with rightwards arrow on top close vertical bar equals square root of 38

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2b
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2 marks

Find the smallest value of a for which

open vertical bar stack O B with rightwards arrow on top close vertical bar greater than open vertical bar stack O A with rightwards arrow on top close vertical bar

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3a
2 marks
Triangle ABC with labelled vertices; A is at the bottom left, B at the right, and C at the top. The triangle is drawn with clear, straight lines.
Figure 1

Figure 1 shows a sketch of triangle A B C.

Given that

  • stack A B with rightwards arrow on top equals negative 3 bold i minus 4 bold j minus 5 bold k

  • stack B C with rightwards arrow on top equals bold i plus bold j plus 4 bold k

find stack A C with rightwards arrow on top.

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3b
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3 marks

Show that cos open parentheses A B C close parentheses equals 9 over 10.

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4
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3 marks

Relative to a fixed origin O, the points A and B have coordinates \left(-2, 5, -7\right) and \left(-7, k, 3\right) respectively, where k is a constant.

Given that the distance AB is 5\sqrt{14}, find the possible values of k.

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5
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5 marks

Figure 2 shows a sketch of triangle ABC.

q1-11-1-vectors-in-2-dimensions-hard-a-level-maths-pure
Figure 2

Given that

\overrightarrow{AB} = -2\mathbf{i} + 3\mathbf{j} - \mathbf{k}

\overrightarrow{AC} = -5\mathbf{i} - 4\mathbf{j} - 7\mathbf{k}

show that the size of angle BAC is 81.9^{\circ} to one decimal place.

How did you do?

6a
3 marks

Relative to a fixed origin O, the point R has coordinates \left(-1, 5, 14\right) and the point S has coordinates \left(7, -2, 12\right).

Find:

(i) the vector \overrightarrow{RS},

(ii) a unit vector in the direction of \overrightarrow{RS}.

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6b
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2 marks

Find the angle that \overrightarrow{RS} makes with the positive y-axis. Give your answer in degrees to one decimal place.

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6c
2 marks

The vector \overrightarrow{TU} is given by

\overrightarrow{TU} = -24\mathbf{i} + 21\mathbf{j} + 6\mathbf{k}

Explain, giving a reason for your answer, whether the vectors \overrightarrow{RS} and \overrightarrow{TU} are parallel.

How did you do?

7
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4 marks

Relative to a fixed origin O, the points P, Q and R have coordinates \left(12, 3, -3\right), \left(7, -8, k\right) and \left(3, 3, -12\right) respectively, where k is a constant.

Given that triangle PQR is an equilateral triangle, find the value of k.

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8
3 marks

The vectors \mathbf{a} and \mathbf{b} are defined by

\mathbf{a} = -12\mathbf{i} - 7\mathbf{j} + 15\mathbf{k}

\mathbf{b} = 4p\mathbf{i} + (pqr + 2qr - p)\mathbf{j} - pq\mathbf{k}

where p, q and r are scalar constants.

Given that \mathbf{a} = \mathbf{b}, find the value of p, the value of q and the value of r.

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9a
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3 marks

A particle P of mass 0.4\text{ kg} is acted upon by a force \mathbf{F}_1 where

\mathbf{F}_1 = \left(-2\mathbf{i} + 6\mathbf{j} + 10\mathbf{k}\right)\text{ N}

Find:

(i) the acceleration of P while the force acts,

(ii) the magnitude of the acceleration of P, giving your answer to 3 significant figures.

How did you do?

9b
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4 marks

A second force \mathbf{F}_2 with a magnitude of 10\sqrt{3}\text{ N} now acts on P. The resultant of \mathbf{F}_1 and \mathbf{F}_2 is parallel to the vector \mathbf{j} and has a magnitude of 8\text{ N}.

Find \mathbf{F}_2, giving your answer in the form \left(x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\right)\text{ N}.

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10a
2 marks

Three forces, \mathbf{F}_1, \mathbf{F}_2 and \mathbf{F}_3, act on a particle of mass 5\text{ kg}. The forces are given by

\mathbf{F}_1 = \left(3\mathbf{i} - 7\mathbf{j} + p\mathbf{k}\right)\text{ N}

\mathbf{F}_2 = \left(q\mathbf{i} + 3\mathbf{j} - \mathbf{k}\right)\text{ N}

\mathbf{F}_3 = \left(-2\mathbf{i} + r\mathbf{j} - 5\mathbf{k}\right)\text{ N}

where p, q and r are scalar constants.

Under the action of these three forces, the particle is in equilibrium.

Find the value of p, the value of q and the value of r.

How did you do?

10b
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5 marks

The third force is now doubled, so that the three forces acting on the particle are \mathbf{F}_1, \mathbf{F}_2 and 2\mathbf{F}_3.

Find:

(i) the resultant force \mathbf{R} now acting on the particle,

(ii) the acceleration of the particle,

(iii) the magnitude of the acceleration of the particle, giving your answer as an exact value.

How did you do?

11a
2 marks

Figure 1 shows a sketch of a cube with vertices O, A, B, C, D, E, F and G.

q9-11-1-vectors-in-2-dimensions-easy-a-level-maths-pure
Figure 1

Relative to a fixed origin O, the edges OA, OB and OC represent the vectors \mathbf{a}, \mathbf{b} and \mathbf{c} respectively. The vertex E is diagonally opposite O, and the vertex G is diagonally opposite A.

Find the vectors \overrightarrow{OE} and \overrightarrow{AG} in terms of \mathbf{a}, \mathbf{b} and \mathbf{c}.

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11b
2 marks

Let P be a point on the line segment OE and let Q be a point on the line segment AG.

Explain why the position vectors \overrightarrow{OP} and \overrightarrow{OQ} can be expressed in the forms

\overrightarrow{OP} = \lambda\,\overrightarrow{OE}

\overrightarrow{OQ} = \mathbf{a} + \mu\,\overrightarrow{AG}

where \lambda and \mu are scalar constants such that 0 \leq \lambda \leq 1 and 0 \leq \mu \leq 1.

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11c
4 marks

By solving the equation \overrightarrow{OP} = \overrightarrow{OQ}, using your results from (a) and (b), show that the diagonals OE and AG intersect each other, and determine the ratio into which they are cut by their point of intersection.

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12
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3 marks

Relative to a fixed origin O, the points A and B have coordinates \left(-1, 3, 14\right) and \left(2k, 3k, 13\right) respectively, where k is an integer.

Given that the distance AB is \sqrt{163}, find the value of k.

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13a
2 marks

Figure 1 shows a sketch of a cube with vertices O, A, B, C, D, E, F and G.

MViXq8lR_q9-11-1-vectors-in-2-dimensions-easy-a-level-maths-pure
Figure 1

Relative to a fixed origin O, the position vectors of the vertices A, B and C are \mathbf{a}, \mathbf{b} and \mathbf{c} respectively.

Find the position vectors \overrightarrow{OF} and \overrightarrow{OG} in terms of \mathbf{a}, \mathbf{b} and \mathbf{c}.

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13b
3 marks

(i) Explain geometrically why \overrightarrow{GE} = \mathbf{a}.

(ii) Show that \overrightarrow{OF} + \overrightarrow{FE} = \overrightarrow{OG} + \overrightarrow{GE}, and hence state the position vector \overrightarrow{OE}.

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13c
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3 marks

Given that the point A has coordinates \left(5, 0, 0\right), find the exact distance OE.

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14a
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3 marks

Relative to a fixed origin O, the points A and B have position vectors left parenthesis 3 bold i minus 2 bold j plus bold k right parenthesis and left parenthesis 5 bold i plus bold j minus 5 bold k right parenthesis respectively.

Find the magnitude of the vector A B.

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14b
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4 marks

The point C has position vector left parenthesis a bold i plus b bold j plus 2 bold k right parenthesis. Given that A, B and C are collinear, find the values of the constants a and b.

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1a
2 marks

Relative to a fixed origin O,

the point A has position vector open parentheses 2 bold i plus 3 bold j minus 4 bold k close parentheses,

the point B has position vector open parentheses 4 bold i minus 2 bold j plus 3 bold k close parentheses,

and the point C has position vector open parentheses a bold i plus 5 bold j minus 2 bold k close parentheses, where a is a constant and a less than 0.

D is the point such that stack A B with rightwards arrow on top equals stack B D with rightwards arrow on top.

Find the position vector of D.

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1b
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3 marks

Given open vertical bar stack A C with rightwards arrow on top close vertical bar equals 4, find the value of a.

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2a
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2 marks
A parallelogram with vertices labelled P, Q, R, and S.
Figure 3

Figure 3 shows a sketch of a parallelogram P Q R S.

Given that

  • stack space P Q with rightwards arrow on top equals 2 bold i plus 3 bold j minus 4 bold k

  • stack Q R with rightwards arrow on top equals 5 bold i minus 2 bold k

show that parallelogram P Q R S is a rhombus.

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2b
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4 marks

Find the exact area of the rhombus P Q R S.

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3a
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5 marks

Figure 1 shows a sketch of triangle ABC.

7R8LyPS9_q1-11-1-vectors-in-2-dimensions-hard-a-level-maths-pure
Figure 1

Given that

\overrightarrow{AB} = 7\mathbf{i} + \mathbf{j} - \mathbf{k}

\overrightarrow{AC} = -2\mathbf{i} + 5\mathbf{k}

show that the size of angle BAC is 119.6^{\circ} to one decimal place.

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3b
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2 marks

Hence find the area of triangle ABC, giving your answer to 3 significant figures.

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4a
2 marks

Relative to a fixed origin O, the points A, B and C have position vectors

bold a equals 2 bold i minus bold j plus 3 bold k

bold b equals 4 bold i plus 2 bold j minus bold k

bold c equals 3 bold i plus 5 bold k

Find the vector A B.

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4b
3 marks

The point D is such that A B C D is a parallelogram.

Find the position vector of D.

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4c
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3 marks

Find the size of angle A B C, giving your answer in degrees to 1 decimal place.

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5a
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3 marks

Relative to a fixed origin O, the point R has position vector \mathbf{i} + 6\mathbf{j} - 2\mathbf{k} and the point S has position vector 10\mathbf{i} + 13\mathbf{k}.

Find a unit vector in the direction of \overrightarrow{RS}.

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5b
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2 marks

Find the angle that \overrightarrow{RS} makes with the negative z-axis. Give your answer in degrees to one decimal place.

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5c
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2 marks

The vector \overrightarrow{TU} is given by

\overrightarrow{TU} = -12\mathbf{i} + 8\mathbf{j} - 20\mathbf{k}

Explain, giving a reason for your answer, whether the vectors \overrightarrow{RS} and \overrightarrow{TU} are parallel.

How did you do?

6
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4 marks

Relative to a fixed origin O, the points P, Q and R have coordinates \left(2, -3, 1\right), \left(-1, -4, 3\right) and \left(k, 0, 3\right) respectively, where k is a constant.

Given that triangle PQR is isosceles, and that k > 1, find the value of k.

How did you do?

7
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4 marks

The vectors \mathbf{a} and \mathbf{b} are defined by

\mathbf{a} = (p + 1)\mathbf{i} - 7\mathbf{j} + (q - 3p)\mathbf{k}

\mathbf{b} = 5\mathbf{i} + (14q + r)\mathbf{j} + (1 - 2r)\mathbf{k}

where p, q and r are scalar constants.

Given that \mathbf{a} = \mathbf{b}, find the value of p, the value of q and the value of r.

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8a
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3 marks

A particle P of mass 0.5\text{ kg} is acted upon by a force \mathbf{F}_1 where

\mathbf{F}_1 = \left(12\mathbf{i} - 4\mathbf{j} + p\mathbf{k}\right)\text{ N}

and p is a scalar constant.

Given that the magnitude of the acceleration of P is 26\text{ m s}^{-2}, find the possible values of p.

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8b
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2 marks

An additional second force \mathbf{F}_2 = q\mathbf{k}\text{ N}, where q is a constant and q < 0, now acts on P. Under the action of the resultant of these two forces, P now experiences an acceleration of magnitude 8\sqrt{10}\text{ m s}^{-2}.

Explain why this additional information shows that p > 0.

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8c
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3 marks

Hence find the value of q.

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9a
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4 marks

Three forces, \mathbf{F}_1, \mathbf{F}_2 and \mathbf{F}_3, act on a particle of mass 5\text{ kg}. The forces are given by

\mathbf{F}_1 = \left(r\mathbf{i} + 5\mathbf{j} + (r - p)\mathbf{k}\right)\text{ N}

\mathbf{F}_2 = \left((p + q)\mathbf{i} - 3p\mathbf{j} + 7\mathbf{k}\right)\text{ N}

\mathbf{F}_3 = \left(-\mathbf{i} + r\mathbf{j} - 2q\mathbf{k}\right)\text{ N}

where p, q and r are scalar constants.

Under the action of these three forces, the particle is in equilibrium.

Find the value of p, the value of q and the value of r.

How did you do?

9b
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5 marks

A fourth force, \mathbf{F}_4, is now added to the particle. Under the combined action of the four forces, the particle experiences an acceleration with a magnitude of 2.2\text{ m s}^{-2} in the same direction as the vector \mathbf{i} - \mathbf{j} + 3\mathbf{k}.

Find \mathbf{F}_4, giving your answer in the form \left(x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\right)\text{ N}, where x, y and z are exact values.

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10
9 marks

Figure 2 shows a sketch of a cube with vertices O, A, B, C, D, E, F and G.

q9-11-1-vectors-in-2-dimensions-easy-a-level-maths-pure
Figure 2

Relative to a fixed origin O, the position vectors of the vertices A, B and C are \mathbf{a}, \mathbf{b} and \mathbf{c} respectively. The face of the cube containing OA and OB is OADB. The face of the cube containing OA and OC is OAFC.

Using vector methods, prove that the diagonals CD and BF bisect each other.

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11
3 marks

Relative to a fixed origin O, the points A and B have coordinates \left(5, 0, -1\right) and \left(k, -2, 3k\right) respectively, where k is a constant.

Given that the distance AB is 6\sqrt{k}, and that the point B lies at a distance of \sqrt{14} from O, find the value of k.

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12
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5 marks

The vectors \mathbf{a} and \mathbf{b} are defined by

\mathbf{a} = pq\mathbf{i} - 24\mathbf{j} + 9r\mathbf{k}

\mathbf{b} = 6\mathbf{i} + (p - r)\mathbf{j} + (p - 3q)\mathbf{k}

where p, q and r are scalar constants.

Given that \mathbf{a} = 3\mathbf{b}, and that r > 0, find the value of p, the value of q and the value of r.

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1a
3 marks

Relative to a fixed origin O

  • the point A has position vector 4 bold i minus 3 bold j plus 5 bold k

  • the point B has position vector 4 bold j plus 6 bold k

  • the point C has position vector negative 16 bold i plus p bold j plus 10 bold k

where p is a constant.

Given that A, B and C lie on a straight line, find the value of p.

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1b
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3 marks

The line segment O B is extended to a point D so that stack C D with rightwards arrow on top is parallel to stack O A with rightwards arrow on top

Find open vertical bar stack O D with rightwards arrow on top close vertical bar , writing your answer as a fully simplified surd.

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2
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8 marks

Figure 1 shows a sketch of a parallelogram ABCD.

q2-11-2-vectors-in-3-dimensions-vh-a-level-maths-pure
Figure 1

Given that

\overrightarrow{AB} = 3\mathbf{i} - \mathbf{j} - 2\mathbf{k}

\overrightarrow{AD} = 7\mathbf{i} - \mathbf{j} + 4\mathbf{k}

find the area of the parallelogram ABCD. Give your answer to 3 significant figures.

How did you do?

3a
5 marks

The vector \overrightarrow{RS} is given by

\overrightarrow{RS} = x\mathbf{i} - 9\mathbf{j} + 3\mathbf{k}

where x is a constant.

The vector \overrightarrow{RS} makes an angle \theta with the positive x-axis.

Show that x^2 = \dfrac{90}{\text{tan}^2 \theta}.

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3b
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4 marks

Given further that \theta is acute and that \text{cos} \; \theta = \dfrac{4}{5}, find a unit vector in the direction of \overrightarrow{RS}.

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4
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6 marks

Figure 2 shows a sketch of a regular tetrahedron.

q4-11-2-vectors-in-3-dimensions-vh-a-level-maths-pure
Figure 2

Relative to a fixed origin O, the points A, B, C and D have coordinates \left(1, 1, 1\right), \left(-8, 10, 1\right), \left(-3, 6, -10\right) and \left(-2k, 13, k\right) respectively, where k is a constant.

Find the coordinates of the point D.

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5a
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3 marks

A particle P of mass 0.2\text{ kg} is acted upon by a force \mathbf{F}_1 where

\mathbf{F}_1 = \left(-3\mathbf{i} + p\mathbf{j} + 4\mathbf{k}\right)\text{ N}

and p is a scalar constant.

Given that the magnitude of the acceleration of P is 65\text{ m s}^{-2}, find the possible values of p.

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5b
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5 marks

An additional second force \mathbf{F}_2 = q\mathbf{j}\text{ N}, where q is a constant and q > 0, now acts on P. Under the action of the resultant of these two forces, P experiences an acceleration of magnitude \dfrac{5\sqrt{221}}{2}\text{ m s}^{-2}.

Find the possible values of q.

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6a
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5 marks

[In this question, the unit vectors \mathbf{i} and \mathbf{j} are horizontal unit vectors directed due east and due north respectively, and \mathbf{k} is a unit vector directed vertically upwards.]

A robotic submarine of mass 750\text{ kg} is initially moving in a level direction such that the \mathbf{k} component of its velocity is zero.

In addition to its weight, the forces acting on the submarine are the combined thrust and lift \mathbf{T}, its buoyancy \mathbf{B}, and the water resistance \mathbf{W}. These forces, measured in newtons, are given by

\mathbf{T} = 600\mathbf{i} - 750\mathbf{j} - 120\mathbf{k}

\mathbf{B} = 7360\mathbf{k}

\mathbf{W} = -500\mathbf{i} + 600\mathbf{j} + 50\mathbf{k}

Taking g = 9.8\text{ m s}^{-2}, find the magnitude of the acceleration of the submarine. Give your answer to 3 significant figures.

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6b
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3 marks

Determine whether the submarine is rising or sinking, giving a reason for your answer, and find the angle its acceleration makes with the vector \mathbf{k}. Give your angle in degrees to one decimal place.

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7
11 marks

Figure 3 shows a sketch of a cuboid with vertices O, A, B, C, D, E, F and G.

q8-11-2-vectors-in-3-dimensions-vh-a-level-maths-pure
Figure 3

Relative to a fixed origin O, the position vectors of the vertices A, B and C are \mathbf{p}, \mathbf{q} and \mathbf{r} respectively. The face of the cuboid containing OA and OC is OAFC. The vertex E is diagonally opposite O.

The point P lies on the space diagonal OE such that it divides OE in the ratio a : b, where a and b are positive constants with a > b.

Using vector methods, show that if the line segment BP is extended, it will intersect the edge FE, and show that FE is divided in the ratio (a - b) : b by the point of intersection.

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