1.13 J: Vectors

Relative to a fixed origin $O$ , the lines $l_1$ and $l_2$ are given by the equations

l_1: \mathbf{r} = (2\mathbf{i} + p\mathbf{j} + 5\mathbf{k}) + \lambda(3\mathbf{i} - \mathbf{j} + 2\mathbf{k})

l_2: \mathbf{r} = (1\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}) + \mu(2\mathbf{i} + 4\mathbf{j} + \mathbf{k})

where $λ\lambda$ and $μ\mu$ are scalar parameters and $p$ is a constant.

Given that $l_1$ and $l_2$ intersect,

find the value of $p$ .

[3]

find the position vector of the point of intersection.

[2]

Find the acute angle between $l_1$ and $l_2$ . Give your answer in degrees to one decimal place.

[3]

The point $A$ lies on $l_1$ with parameter $λ=1\lambda = 1$ . The point $B$ lies on $l_2$ with $AB⃗\vec{AB}$ perpendicular to $l_2$ .

Find the coordinates of $B$ .

[4]