DP Mathematics: Analysis and Approaches Questionbank

Given that $L$ meets ${\Pi }_1$ at the point $\text{P}$, find the coordinates of $\text{P}$.

Find the shortest distance from the point $\text{O}(0, 0, 0)$ to ${\Pi }_1$.

Find the equation of ${\Pi }_2$, giving your answer in the form $\mathit{r}\mathbf{.}\mathit{n}=d$.

Determine the acute angle between ${\Pi }_1$ and ${\Pi }_2$.

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 $\text{ABCD}$.

Find a Cartesian equation of the plane ${\text{Π}}_3$ which is perpendicular to ${\text{Π}}_1$ and ${\text{Π}}_2$ and passes through the origin $(0, 0, 0)$.

Find the coordinates of the point where ${\text{Π}}_1$, ${\text{Π}}_2$ and ${\text{Π}}_3$ intersect.

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 ${\Pi }_3$.

Verify that the point $\text{P}(1, -2, 0)$ lies on both $\prod _1$ and $\prod _2$.

Find a vector equation of $L$, the line of intersection of $\prod _1$ and $\prod _2$.

Show that the three planes do not intersect.

Find the distance between $L$ and $\prod _3$.

Given that a $\times$ b $=$ b $\times$ c $\ne$ 0 prove that a $+$ c $=$ sb where s is a scalar.

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.

Find how many sets of four points can be selected which can form the vertices of a quadrilateral.

Find how many sets of three points can be selected which can form the vertices of a triangle.

Verify that $\text{P}$ is the point of intersection of the two lines.

Write down the value of $\lambda$ corresponding to the point $\text{A}$.

Write down $\overrightarrow{\text{PA}}$ and $\overrightarrow{\text{PB}}$.

Let $\text{C}$ be the point on $l_1$ with coordinates (1, 0, 1) and $\text{D}$ be the point on $l_2$ with parameter $\mu =-2$.

Find the area of the quadrilateral $\text{CDBA}$.

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

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

Hence or otherwise, find the area of the triangle ABC.

Find a $\times$ b.

Hence find the Cartesian equation of the plane containing the vectors a and b, and passing through the point $(1,0,-1)$.

Show that $\overrightarrow{\text{AB}}\times \overrightarrow{\text{AV}}=-10\left(\begin{array}{c}10-2p\\ p\\ p\end{array}\right)$ and find a similar expression for $\overrightarrow{\text{AC}}\times \overrightarrow{\text{AV}}$.

Hence, show that, if the angle between the faces $\text{ABV}$ and $\text{ACV}$ is $\theta$, then $\text{cos}\theta =\frac{p\left(3p-20\right)}{6p^2-40p+100}$.

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

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

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 $\text{AOC}$.