DP Mathematics: Analysis and Approaches Questionbank

Show that the point $(-1, 0, 3)$ lies on $L_1$.

Find a vector equation of $L_1$.

Find the possible values of $a$ when the acute angle between $L_1$ and $L_2$ is $45°$.

It is given that the lines $L_1$ and $L_2$ have a unique point of intersection, $\text{A}$, when $a\ne k$.

Find the value of $k$, and find the coordinates of the point $\text{A}$ in terms of $a$.

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

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

Find the coordinates of $\text{P}$.

Determine the length of time between the first airplane arriving at $\text{P}$ and the second airplane arriving at $\text{P}$.

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

Show that the three planes do not intersect.

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

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

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${}_3$.

Show that the two submarines would collide at a point P and write down the coordinates of P.

Show that submarine B travels in the same direction as originally planned.

Find the value of t when submarine B passes through P.

Find an expression for the distance between the two submarines in terms of t.

Find the value of t when the two submarines are closest together.

Find the distance between the two submarines at this time.

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.