Review practice · Section 1 of 2

The line L has equation $\mathbf r=(1,3,7)+t(2,-1,5)$, $t\in\mathbb R$. A,B have position vectors $(9,3,26)$ and $(13,9,\alpha)$. L intersects the line through A,B at P. Find α and the acute angle between L and AB.7 marks

Line l passes through A=(1,2,−1) and B=(2,a,1).

(i) Show that l cuts the x-axis when a=−2.3 marks

(ii) Find the length of projection of OA onto l and hence the shortest distance from the origin to l.5 marks

Relative to O, A,B,C have position vectors $\mathbf i+6\mathbf j-2\mathbf k$, $-3\mathbf i+2\mathbf j-5\mathbf k$, $3\mathbf i-2\mathbf j+2\mathbf k$. L passes through A,C.

(i) Find a vector equation of L.2 marks

(ii) Find the exact length of projection of AB onto L.3 marks

(iii) Hence or otherwise find the shortest distance from B to L in exact form.2 marks

D is such that ABCD is a parallelogram. Use a vector product to find its exact area.3 marks

Relative to O, A,B,C have position vectors $5\mathbf i+4\mathbf j+10\mathbf k$, $-4\mathbf i+4\mathbf j-2\mathbf k$, $-5\mathbf i+9\mathbf j+5\mathbf k$.

(i) Find the Cartesian equation of AB.3 marks

(ii) Find the length of projection of AC onto AB.3 marks

(iii) Hence or otherwise find the perpendicular distance from C to AB and the position vector of its perpendicular foot N.6 marks

(iv) D lies on CN produced such that N is the midpoint of CD. Find the position vector of D.2 marks

Relative to O, A,B,C have position vectors $-3\mathbf i+8\mathbf j+\mathbf k$, $7\mathbf i+3\mathbf j+6\mathbf k$, and $\alpha\mathbf i+\beta\mathbf j-6\mathbf k$ respectively, where α,β are real.

(i) Given $BC:\mathbf r=(7+3\lambda)\mathbf i+3\mathbf j+(6+4\lambda)\mathbf k$, find α,β.2 marks

(ii) Find the length of projection of AB onto BC.2 marks

(iii) Hence or otherwise find the position vector of the perpendicular foot from A to BC.3 marks

(iv) Find the exact area of triangle ABC.3 marks

Relative to O, A,B have position vectors $4\mathbf i+3\mathbf j$ and $5\mathbf i+7\mathbf j$. Line $l_1$ passes through B parallel to $\mathbf i+2\mathbf j+\alpha\mathbf k$. Line $l_2$ has Cartesian equation $(1-x)/2=(z-2)/1$, y=4.

(i) Given $l_1,l_2$ are perpendicular, show α=2.2 marks

(ii) Find the length of projection of BA onto $l_1$. Hence find vector BN, where N is the perpendicular foot from A onto $l_1$.4 marks

(iii) A′ is the reflection of A in $l_1$, and C makes AA′CB a parallelogram. Find its exact area.3 marks