Vector Properties (DP IB Analysis & Approaches (AA): HL): Exam Questions

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

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

When the angle between $\mathit{a}$ and $\mathit{c}$ is $\frac{\pi }{4}$, show that $\left|\mathit{a}·\left(\mathit{b}+\mathit{c}\right)\right|=16+20\sqrt{2}$.

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 $\mathit{v}=\left(\begin{array}{c}t-6\\ 4\\ t-4\end{array}\right)$. 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 $\mathrm{C}\hat{\mathrm{D}}\mathrm{A}$ .

$\mathrm{ABCDEFGH}$ is a cuboid as shown in the diagram below.

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

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

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

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

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

The points $\mathrm{A},\mathrm{B},\mathrm{C}\mathrm{and}\mathrm{D}$ form the vertices of a parallelogram with position vectors $\mathit{a},\mathit{b},\mathit{c}\mathrm{and}\mathit{d}$  respectively.

Show that the area of the parallelogram is $\left|\mathit{a}\times \mathit{b}+\mathit{b}\times \mathit{d}+\mathit{d}\times \mathit{a}\right|$.

Hence show that the shortest distance from $\mathrm{B}$ to $\mathrm{AD}$ is $\frac{\left|\mathit{a}\times \mathit{b}+\mathit{b}\times \mathit{d}+\mathit{d}\times \mathit{a}\right|}{\left|\mathit{d}-\mathit{a}\right|}$ .

Point $\mathrm{A}$  has position vector $\mathit{a}=3\mathrm{i}+2\mathrm{j}-\mathrm{k}$ and point B has position vector $\mathit{b}=\mathrm{i}-\mathrm{j}+4\mathrm{k}$  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 $\mathrm{A}(5,7,-3),\mathrm{B}(6,10,-2),\mathrm{C}(9,11,0)\mathrm{and}\mathrm{E}(4,5,4)$.

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

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

Given that the angle between $a$ and $c$ is $\frac{\pi }{3}$, find $\left|a\times \left(b+c\right)\right|$ .

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

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

Find $\cos \theta$.

A third point $C$ is located such that its position vector  $c=\left(\begin{array}{c}28\\ -44\\ 32\end{array}\right)$

Show that $c=4a\times b$.

$\mathrm{ABCDE}$ 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 $\left|\mathrm{HX}\right|$ .

Show that for any two vectors $a$ and $b$,${\left|a\times b\right|}^2-{\left(a·b\right)}^2={\left|a\right|}^2{\left|b\right|}^2\cos 2\theta$

Hence show that when $\sin \theta =\frac{\sqrt{2}}{2},\left|a\times b\right|=a·b$.

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.