Vectors

Show that the lines are skew.

Find the obtuse angle between the directions of the two lines.

Find a vector equation for the line $l_1$.

The line $l_2$ has equation $\mathbf{r} = 2\mathbf{i} + \mathbf{j} + 5\mathbf{k} + \mu(\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})$.

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

Find the acute angle between the directions of $l_1$ and $l_2$.

Find a vector equation for the line through $A$ and $B$.

Using a scalar product, find the exact value of $\cos BAC$.

Hence find the exact area of triangle $ABC$.

Find a vector equation for $l$.

The point $P$ is the foot of the perpendicular from $A$ to $l$.

Find the position vector of $P$.

The point $D$ is the reflection of $A$ in $l$.

Find the position vector of $D$.

It is given that $|\vec{AB}| = |\vec{BC}|$.

Find the value of $b$.

$A$, $B$, $C$ and $D$ are the vertices of a rhombus.

Find the position vector of $D$.

Calculate angle $ABC$.

Find a vector equation for $m$.

Find the position vector of the point of intersection of $m$ and the line passing through the points $C$ and $D$.

Find the position vector of the foot of the perpendicular from $C$ to $m$.

Find the vectors $\vec{MB}$ and $\vec{MC}$.

Calculate the exact value of the cosine of angle $CMB$.

Hence or otherwise find the exact area of triangle $ABC$.

Write down a vector equation for $l_1$ and find a vector equation for $l_2$.

Calculate the acute angle between $l_1$ and $l_2$.

Find the position vector of the point of intersection of $l_1$ and $l_2$.

Find the value of $a$ for which $l_1$ is perpendicular to $l_2$.

Find the value of $a$ for which $l_1$ and $l_2$ intersect.

Find the values of $a$ for which the acute angle between $l_1$ and $l_2$ is equal to $\cos^{-1}(\frac{5}{14})$.

Show that $OABC$ is a rectangle.

Use a scalar product to find the acute angle between the diagonals of $OABC$.