Vector Properties (DP IB Applications & Interpretation (AI): HL): Exam Questions

Given $\left|a\right|=5$, $b=\left(\begin{array}{c}12\\ -5\end{array}\right)$  and $a·b=16$  ,  find the angle between $a$ and $b$.

Consider a third vector c, where $\left|c\right|=8.$

When the angle between $a$ and $c$ is  $\frac{\pi }{4}$ , 

(i) verify that $\left|a\right|\left|c\right|<2\left|a·c\right|,$

(ii) find the component of vector $c$ acting in the direction of vector $a.$

The points $\mathrm{A}$ and $\mathrm{B}$  have position vectors $a$  and $b$ respectively.

 $\left|\mathit{a}\right|=9,\mathit{a}\mathbf{\times }\mathit{b}=\left(\begin{array}{c}5\\ -3\\ 2\end{array}\right),\mathit{a}\mathbf{·}\mathit{b}=16$ 

Find the angle between $\mathit{a}$ and $\mathit{b}$.

The points $\mathrm{A}$ and $\mathrm{B}$ form a triangle with the origin $\mathrm{O}$.

Find $\left|\mathit{b}\right|$.

 $\mathrm{ABCD}$ is a parallelogram with vertices $\mathrm{A}\left(-2,5,3\right),\mathrm{B},\mathrm{C}$ and $\mathrm{D}(3,-1,t)$ where $t>0$ .

 $\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{DC}}=\left(\begin{array}{c}2\\ 4\\ 1\end{array}\right)$ 

Given that the area of the parallelogram is $\sqrt{1221}$ units, find the value of t.

Two points $\mathrm{A}$ and $\mathrm{B}$  have position vectors $a=\left(\begin{array}{c}6\\ 2\\ -5\end{array}\right)$  and $b=\lambda \left(\begin{array}{c}3\\ -2\\ -1\end{array}\right)$ respectively. 

A third point $C$ is located such that $\overrightarrow{\mathrm{AC}}=\left(\begin{array}{c}8\\ -2\\ 0\end{array}\right)$. 

Given that the angle between the vectors $\overrightarrow{\mathrm{AB}}$  and $\overrightarrow{\mathrm{AC}}$ is obtuse, find the range of possible values for $\lambda$.

In vector form, the two parallel sides of a trapezium are given by $\mathit{u}=\left(\begin{array}{c}t\\ 2t-6\\ 1\end{array}\right)$  and $v=\left(\begin{array}{c}t-6\\ 4\\ t-4\end{array}\right)$. Additionally, $\left|\mathit{u}\right|\left|\mathit{v}\right|=18$.

Given that $t$ is an integer, find the value of $t$.

A third side of the trapezium, with vector $\mathit{w}=\left(\begin{array}{c}x\\ 3\\ z\end{array}\right)$, is perpendicular to both $\mathit{u}$ and $\mathit{v}$.

Given that $\left|\mathit{w}\right|=\sqrt{26}$, and that $x$ is an integer, find the values of $x$ and $z$.

The points $\mathrm{A}(-2,5,3),\mathrm{B},\mathrm{C}\mathrm{and}\mathrm{D}$ form a parallelogram.

$\overrightarrow{\mathrm{AB}}=\left(\begin{array}{c}6\\ 2\\ -1\end{array}\right)$, $\overrightarrow{\mathrm{BC}}=\left(\begin{array}{c}-4\\ 3\\ 4\end{array}\right)$ 

Find the area of the parallelogram.

Show that the diagonals of the parallelogram are perpendicular to each another.

Determine the nature of angle $\mathrm{C}\hat{\mathrm{D}}\mathrm{A}$ .

ABCDEFGH is a cuboid as shown in the diagram below.

Point A is located at $\left(5,-3,4\right),\overrightarrow{AB}=2\mathrm{j}-\mathrm{k}$ and $\overrightarrow{\mathrm{BC}}=4\mathrm{i}+\mathrm{j}+2\mathrm{k}.$

The perpendicular distance between the faces ABCD and EFGH of the cuboid is $2\sqrt{105}$  units.

Find the coordinates of the point $\mathrm{E}\left(x,y,z\right)$, where $x,y,z\in {\mathrm{\mathbb{Z}}}^{+}$.

A triangle is formed inside the cuboid by connecting the vertices B, C and E, where $\mathrm{B}\hat{\mathrm{C}}\mathrm{E}=\theta$. 

Using vector methods, find $\cos \theta$ .

The diagram below shows three forces acting on a particle.

Find the magnitude and direction of the resultant forces acting on the particle in the horizontal and vertical directions.

Point $\mathrm{A}$  has position vector $a=3\mathrm{i}+2\mathrm{j}-\mathrm{k}$ and point B has position vector $b=\mathrm{i}-\mathrm{j}+4\mathrm{k}$  relative to the origin $\mathrm{O}$.

Find the area of the triangle $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}$ .

Point $\mathrm{X}$  is located a distance of 8 units from the origin in the direction perpendicular to the plane formed by $\mathrm{AOB}$.

Find all possible vectors $\overrightarrow{\mathrm{OX}}$ .

Find the volume of the tetrahedron $\mathrm{AOBX}$ . Give your answer in the form $c\sqrt{d}$, where $c,d\in \mathrm{\mathbb{Z}}$ .

 $\mathrm{ABCDEFGH}$ are vertices of a parallelepiped with the vectors $\overrightarrow{\mathrm{AB}},\overrightarrow{\mathrm{AD}}\mathrm{and}\overrightarrow{\mathrm{AE}}$ defined as $p,q$and $r$ respectively. $\theta$ is the angle between $\overrightarrow{\mathrm{AE}}$  and the normal to the base ABCD. This information can be seen in the diagram below.

Find an expression for

(i) the area of the base ABCD,

(ii) the perpendicular height of the parallelepiped.

Hence, show that the volume of a parallelepiped is given by $\left|\left(p\times q\right)·r\right|$units³.

Find the volume of a parallelepiped with vertices $\mathrm{A}(5,7,-3),\mathrm{B}(6,10,-2),\mathrm{C}(9,11,0)\mathrm{and}\mathrm{E}(4,5,4)$.

A hot air balloon travels vertically upwards from the point O on the ground for 12 metres and then moves in a straight line with constant velocity described by the vector $v_1=2\mathrm{i}-\mathrm{j}+3\mathrm{k}{\mathrm{ms}}^{-1}$.  At the same time a truck leaves a point O and moves along the ground below the hot air balloon at a velocity of $7m{s}^{-1}$ along a bearing of $079°$. 

All distances are measured in metres and time in seconds. The base vectors $\mathrm{i}$ and $\text{j}$  represent due east and due north respectively and the base vector $\mathrm{k}$ points upwards.

Write down

(i) the speed at which the hot air balloon is travelling,

(ii) the position vector of the hot air balloon five minutes after it changed direction,

(iii) the velocity of the truck in base vector form.

When it reaches a vertical height of 0.92 km the hot air balloon changes direction again and moves with constant velocity described by the vector $v_2=2\mathrm{i}-\mathrm{j}{\mathrm{ms}}^{-1}$. 

Find the component of the balloon’s velocity in the direction of the road the truck had driven along.

Given $\left|\mathit{a}\right|=6,\left|\mathit{b}\right|=2$ and $\mathit{a}\mathbf{\times }\mathit{b}=\left(\begin{array}{c}-5\\ 2\\ 7\end{array}\right)$ find the possible values of  $\mathit{a}\mathbf{·}\mathit{b}$ .

nsider a third vector c, where $\left|c\right|=5$.

Given that the angle between $a$ and $c$ is  $\frac{\mathrm{\pi }}{3}$,

(i) find $\left|a·c\right|$ and $\left|a\times c\right|$,

(ii) find the component of vector $c$ acting perpendicular to vector $a$.

The points A and B have position vectors $a$ and $b$ respectively.

$\left|a\right|=\sqrt{35},b=2\mathrm{i}+2\mathrm{j}+\mathrm{k},a\times b=\left(\begin{array}{c}7\\ -11\\ 8\end{array}\right)$ and $\theta$  is the angle between $a$ and $b$.

Find $\cos \theta$ .

A third point C is located such that its position vector  $c=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$

Given that $c=n\left(a\times b\right)$  find the unit vector in the direction of c.

$\mathrm{ABCDE}$ is a pentagon, where  $\mathrm{A}\left(0,4,-1\right),\mathrm{D}(-1,7,3),\overrightarrow{\mathrm{AB}}=\left(\begin{array}{c}-4\\ 2\\ 5\end{array}\right)$ and $\overrightarrow{\mathrm{CD}}=\left(\begin{array}{c}1\\ -1\\ -1\end{array}\right)$ .

Given that $\overrightarrow{\mathrm{BD}}=\overrightarrow{\mathrm{AE}}$ find the area of triangle $\mathrm{B}\hat{\mathrm{C}}\mathrm{D}$ as a percentage of the total area of the pentagon.

Three points, $\mathrm{A}(-3,-1)$, $\mathrm{B}(2,2)$  and C, are located on a straight line where $\overrightarrow{\mathrm{AC}}=\lambda \overrightarrow{\mathrm{AB}}$. A fourth point $\mathrm{D}$, is located such that $\overrightarrow{\mathrm{DB}}$ is perpendicular to $\overrightarrow{\mathrm{OB}}$ and $\left|\mathrm{DB}\right|=2\sqrt{22}$ .

Find $\mathrm{D}\hat{\mathrm{B}}\mathrm{C}$.

Given that the area of the triangle $\mathrm{BCD}=53.1$units² correct to 3 significant figures, find $\lambda$.

Consider a regular hexagon $\mathrm{ABCDEF}$ with sides of length $9\sqrt{2}$units. The position vectors of $\mathrm{A}$ and $\mathrm{E}$ are $a=-11\mathrm{i}-4\mathrm{j}+5\mathrm{k}$  and $e=3\mathrm{i}-3\mathrm{j}-12\mathrm{k}$ respectively.

Given that the coordinates of $\mathrm{F}$ are $\left(r,t,t\right)$,  where $r,t\in \mathrm{\mathbb{Z}},r\ne 0.$ find the value of $r$ and $t$.

ABCD is a parallelogram defined by the vectors $\overrightarrow{\mathrm{AB}}=p$ and $\overrightarrow{\mathrm{AD}}=q$, where $p=\left(\begin{array}{c}1\\ 5\end{array}\right)$ and $q=\left(\begin{array}{c}2t+2\\ t\end{array}\right)$. 

Given that the angle $\mathrm{B}\hat{\mathrm{A}}\mathrm{D}$ is acute, find the range of values for $t$.

$\overrightarrow{\mathrm{AB}}$ is enlarged by a factor of $k$.

Show that $kp·q=k\left(p·q\right)$.

Given that $k=7$, find the range of possible values for the area of the enlarged parallelogram.

Consider the cuboid $\mathrm{ABCDEFGH}$ as shown in the diagram below. The position vectors of $\mathrm{A},\mathrm{C},\mathrm{D}$ and $\mathrm{E}$  are  $a=-5\mathrm{i}+2\mathrm{j}-4\mathrm{k}$, $c=2\mathrm{i}+3\mathrm{j}+\mathrm{k}$, $d=-\mathrm{i}+\mathrm{j}$ and $e=-3\mathrm{i}+\mathrm{j}+2\mathrm{k}$ respectively.

$\mathrm{X}$ is a point located on the line $\left[\mathrm{EC}\right]$ such that $\overrightarrow{\mathrm{EX}}=\mathrm{\lambda }\overrightarrow{\mathrm{EC}}$.

Find the shortest length $\left|\mathrm{HX}\right|$ .

The following force diagram shows three forces acting on a particle:

Given that the resultant force on the particle in the vertical direction is 14.195 N downwards, find the size of the angle $\theta$.

Consider a tetrahedron $\mathrm{ABCV}$ where $\overrightarrow{\mathrm{AB}}=p$, $\overrightarrow{\mathrm{AC}}=q$ and $\overrightarrow{\mathrm{AV}}=r$. The perpendicular height, $h$ , of the tetrahedron from the base $\mathrm{ABC}$ makes an angle of $\theta$ with $r$.

This information is shown in the diagram below.

Find an expression for the volume of the tetrahedron in terms of $p,qandr.$

Find the volume of the tetrahedron when $p=\left(\begin{array}{c}4\\ 5\\ -2\end{array}\right)$, $q=\left(\begin{array}{c}2\\ 2\\ -5\end{array}\right)$, $r=\left(\begin{array}{c}1\\ 3\\ 4\end{array}\right)$

Hence find the shortest distance between vertex A and its opposite face.

Consider a parallelepiped $\mathrm{ABCDEFGH}$  with vertices $\mathrm{A}\left(2,5,-1\right),\mathrm{B}\left(3,8,0\right),\mathrm{C}\left(5,4,3\right)$ and $\mathrm{H}(5,2,7)$  as seen in the diagram below.

By first finding an expression for the perpendicular height of the object, find the volume of the parallelepiped.

An explorer is trying to find some treasure that they believe is hidden in a wild region near an underground river. They set off from a point O and travel at a velocity of $4{\mathrm{kmh}}^{-1}$along a bearing of $229°$. The river flows in the direction given by the vector $\left(\begin{array}{c}1\\ -5\end{array}\right)$. 

The base vectors $\mathrm{i}\mathrm{and}\mathrm{j}$ represent due east and due north respectively. 

Write down the velocity of the explorer as a column vector.

Find the component of the explorer’s velocity in the direction of the underground river.

After they have travelled for 2.5 hours, the explorer finds the treasure.

Find the perpendicular distance between the location of the treasure and the underground river.