DP Mathematics: Analysis and Approaches Questionbank

Show that $l_1$ and $l_2$ are never perpendicular to each other.

Find the vectors $\overrightarrow{\text{AB}}$ and $\overrightarrow{\text{AC}}$.

Use a vector method to show that $\text{B}\hat{\text{A}}\text{C}=60°$.

Show that the Cartesian equation of the plane $\Pi$ that contains the triangle $\text{ABC}$ is $-x+y+z=-2$.

Find a vector equation of the line $L$.

Hence determine the minimum distance, $d_{\text{min}}$, from $\text{D}$ to $\Pi$.

Find the volume of right-pyramid $\text{ABCD}$.

Given any two non-zero vectors, $\mathit{a}$ and $\mathit{b}$, show that ${\left|\mathit{a}\times \mathit{b}\right|}^2={\left|\mathit{a}\right|}^2{\left|\mathit{b}\right|}^2-{\left(\mathit{a}·\mathit{b}\right)}^2$.

Show that $l_1$ and $l_2$ do not intersect.

Find the minimum distance between $l_1$ and $l_2$.

Find the vector $\overrightarrow{\text{AB}}$ and the vector $\overrightarrow{\text{AC}}$.

Hence find the equation of ${\Pi }_1$, expressing your answer in the form $ax+by+cz=d$, where $a, b, c, d\in \mathrm{\mathbb{Z}}$.

Show that at the point $\text{P}, \lambda =\frac{3}{4}$.

Find the reflection of the point $\text{B}$ in the plane ${\Pi }_3$.

The line $L$ is the intersection of ${\Pi }_1$ and ${\Pi }_2$. Verify that the vector equation of $L$ can be written as $\mathit{r}=\left(\begin{array}{c}0\\ -2\\ 0\end{array}\right)+\lambda \left(\begin{array}{c}1\\ 1\\ -1\end{array}\right)$.

Hence find the coordinates of $\text{P}$.

Hence find the vector equation of the line formed when $L$ is reflected in the plane ${\Pi }_3$.

Find $\mathit{q}$ such that $\left|\mathit{q}\right|=\left|\mathit{b}\right|$ and $\mathit{q}$ is perpendicular to $\mathit{a}$.

Find the acute angle between the two airplanes’ lines of flight. Give your answer in degrees.

Let $D\left(t\right)$ represent the distance between airplane $A$ and airplane $B$ for $0\le t\le 2.5$.

Find the minimum value of $D\left(t\right)$.

Find the area of the parallelogram ABCD.

By using a suitable scalar product of two vectors, determine whether $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ is acute or obtuse.

The acute angle between the vectors 3i − 4j − 5k and 5i − 4j + 3k is denoted by θ.

Find cos θ.

Find the vector equation of the line (BC).

Determine whether or not the lines (OA) and (BC) intersect.

Find the Cartesian equation of the plane Π${}_1$, which passes through C and is perpendicular to $\overrightarrow{\mathrm{O}\mathrm{A}}$.

Show that the line (BC) lies in the plane Π${}_1$.

Verify that 2j + k is perpendicular to the plane Π${}_2$.

Find a vector perpendicular to the plane Π${}_3$.

Find the acute angle between the planes Π${}_2$ and Π${}_3$.

Find and simplify an expression for a • b in terms of $t$.

Hence or otherwise, find the values of $t$ for which the angle between a and b is obtuse .

Let a = $\left(\begin{array}{c}2\\ k\\ -1\end{array}\right)$ and b = $\left(\begin{array}{c}-3\\ k+2\\ k\end{array}\right)$, $k\in \mathbb{R}$.

Given that a and b are perpendicular, find the possible values of $k$.

Find $\left|\overrightarrow{\text{AB}}\right|$.

Let $\overrightarrow{\text{AC}}=\left(\begin{array}{c}3\\ 0\\ 0\end{array}\right)$. Find $\mathrm{B}\hat{\mathrm{A}}\mathrm{C}$.

Show that $\overrightarrow{\text{AB}}=\left(\begin{array}{c}2\\ -1\\ 1\end{array}\right)$.

Find a vector equation for $L$.

Find the value of $p$.

The point D has coordinates $(q^2,0,q)$. Given that $\overrightarrow{\text{DC}}$ is perpendicular to $L$, find the possible values of $q$.

Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.
This is shown in the following diagram.

The vectors p , q and r are shown on the diagram.

Find p•(p + q + r).

Show that $\overrightarrow{\text{AB}}=\left(\begin{array}{c}6\\ 8\\ -5\end{array}\right)$

Find a vector equation for L.

Point C (k , 12 , −k) is on L. Show that k = 14.

Find $\overrightarrow{\text{OB}}\bullet \overrightarrow{\text{AB}}$.

Write down the value of angle OBA.

Point D is also on L and has coordinates (8, 4, −9).

Find the area of triangle OCD.

Find $\overrightarrow{\text{PQ}}$.

Find $\left|\overrightarrow{\text{PQ}}\right|$.

Find the angle between PQ and PR.

Find the area of triangle PQR.

Hence or otherwise find the shortest distance from R to the line through P and Q.

Find the value of $k$.

Given that c = a + 2b, find c.

Find $\overrightarrow{AB}$.

Hence, write down a vector equation for $L_1$.

A second line $L_2$, has equation r = $\left(\begin{array}{c}1\\ 13\\ -14\end{array}\right)+s\left(\begin{array}{c}p\\ 0\\ 1\end{array}\right)$.

Given that $L_1$ and $L_2$ are perpendicular, show that $p=2$.

The lines $L_1$ and $L_1$ intersect at $C(9,13,z)$. Find $z$.

Find a unit vector in the direction of $L_2$.

Hence or otherwise, find one point on $L_2$ which is $\sqrt{5}$ units from C.

The vector equation of line $L$ is given by r $=\left(\begin{array}{c}-1\\ 3\\ 8\end{array}\right)+t\left(\begin{array}{c}4\\ 5\\ -1\end{array}\right)$.

Point P is the point on $L$ that is closest to the origin. Find the coordinates of P.

Find a vector equation for L₁.

The vector $\left(\begin{array}{c}2\\ p\\ 0\end{array}\right)$ is perpendicular to $\overrightarrow{\text{AB}}$. Find the value of p.

Find $\overrightarrow{\text{AB}}$.

Find $\left|\overrightarrow{\text{AB}}\right|$.

Find the value of $y$.

Show that $\overrightarrow{\text{AC}}=\left(\begin{array}{c}8\\ -10\\ -1\end{array}\right)$.

Find the angle between $\overrightarrow{\text{AB}}$ and $\overrightarrow{\text{AC}}$.

Find the area of triangle ABC.

Find a vector equation of the line that passes through P and Q.

The line through P and Q is perpendicular to the vector 2i $+$ nk. Find the value of $n$.

The magnitudes of two vectors, u and v, are 4 and $\sqrt{3}$ respectively. The angle between u and v is $\frac{\pi }{6}$.

Let w = u − v. Find the magnitude of w.

parallel.

perpendicular.

$\overrightarrow{\text{OA}}\bullet \overrightarrow{\text{OC}}$.

$\overrightarrow{\text{OB}}\bullet \overrightarrow{\text{OC}}$.

Given that $\text{A}\hat{\text{O}}\text{C}=\text{B}\hat{\text{O}}\text{C}$, show that $k=7$.

Calculate the area of triangle $\text{AOC}$.

Consider the vectors a = $\left(\begin{array}{c}3\\ 2p\end{array}\right)$ and b = $\left(\begin{array}{c}p+1\\ 8\end{array}\right)$.

Find the possible values of p for which a and b are parallel.