Practice section 1 of 9

Alice walks 5 units in the direction of $\mathbf p$. She then walks 3 units in the direction of $\mathbf q$. What is the expression of the resultant vector with reference to her starting point? ($\mathbf p$ and $\mathbf q$ are non-parallel vectors.)

(a) $5\widehat{\mathbf p}+3\widehat{\mathbf q}$; (b) $5\mathbf p+3\mathbf q$; (c) no difference.

A point $A$ has coordinates $(1,2,3)$. Find (i) $\overrightarrow{OA}$, (ii) $\lvert\overrightarrow{OA}\rvert$, (iii) the unit direction vector in the direction of $\overrightarrow{OA}$, and (iv) a vector with magnitude 4 units in that direction.

Show that the line segment joining the midpoint of two sides of a triangle is parallel to the third side and half its length.

$O,A,B,C$ are four distinct points with $\overrightarrow{OA}=10\mathbf a$, $\overrightarrow{OB}=5\mathbf b$ and $\overrightarrow{OC}=4\mathbf a+3\mathbf b$. Show that $A,B,C$ are collinear.

Given $A(1,2,2)$, $B(4,5,7)$, $C(7,8,12)$ and $D(1,2,4)$: (i) show that A,B,C are collinear; (ii) show that A,B,D are not collinear.

The position vectors of $A,B,C,D$ are $\begin{pmatrix}-1\\4\\-3\end{pmatrix}$, $\begin{pmatrix}1\\5\\-2\end{pmatrix}$, $\begin{pmatrix}5\\7\\0\end{pmatrix}$ and $\begin{pmatrix}2\\1\\1\end{pmatrix}$ respectively. Evaluate (i) $\overrightarrow{AB}$, (ii) $\overrightarrow{BC}$, (iii) $2\overrightarrow{AB}$, (iv) $\overrightarrow{DB}$. Hence or otherwise determine (v) whether A,B,C are collinear and (vi) the shape of O,A,B,D.