DP Mathematics: Analysis and Approaches Questionbank

the value of $m$.

the condition on the value of $p$.

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.

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 $\prod _3$.

Find the Cartesian equation of plane Π containing the points $\text{A}\left(6,2,1\right)$ and $\text{B}\left(3,-1,1\right)$ and perpendicular to the plane $x+2y-z-6=0$.

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 the Cartesian equation of the plane ${\mathrm{\Pi }}_1$, passing through the points A , B and D.

Find the angle between the faces ABD and BCD.

Find the Cartesian equation of ${\mathrm{\Pi }}_3$.

Show that P is the midpoint of AD.

Find the area of the triangle OPQ.

Explain why ABCD is a parallelogram.

Using vector algebra, show that $\overrightarrow{\text{AD}}=\overrightarrow{\text{BC}}$.

Show that p = 1, q = 1 and r = 4.

Find the area of the parallelogram ABCD.

Find the vector equation of the straight line passing through M and normal to the plane $\mathrm{\Pi }$ containing ABCD.

Find the Cartesian equation of $\mathrm{\Pi }$.

Find the coordinates of X, Y and Z.

Find YZ.

Find the Cartesian equation of the plane containing P, Q and R.

Given that П₁ and П₂ meet in a line $L$, verify that the vector equation of $L$ can be given by r $=\left(\begin{array}{c}\frac{5}{4}\\ 0\\ -\frac{7}{4}\end{array}\right)+\lambda \left(\begin{array}{c}\frac{1}{2}\\ 1\\ -\frac{5}{2}\end{array}\right)$.

Given that П₃ is parallel to the line $L$, show that $a+2b-5c=0$.

Show that $5a-7c=4$.

Given that П₃ is equally inclined to both П₁ and П₂, determine two distinct possible Cartesian equations for П₃.

Find, in terms of $b$, a Cartesian equation of the plane Π containing this triangle.

Find, in terms of $b$, the equation of the line L which passes through M and is perpendicular to the plane П.

Show that L does not intersect the $y$-axis for any negative value of $b$.

Find a $\times$ b.

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