Practice section 3 of 6

Find a vector equation and cartesian equation of the following lines:

(a) passing through the point with position vector $7\mathbf i+2\mathbf j-4\mathbf k$ and parallel to to $\mathbf i-3\mathbf j+\mathbf k$

(b) passing through the points $(2,-2,1)$ and $(0,4,9)$

(c) passing through the point with position vector $7\mathbf i$ and parallel to the line $\mathbf r=(2-p)\mathbf i+p\mathbf j+5\mathbf k$.

Convert the equations of following lines to their vector equation form:

(a) $3y=2x-6=\dfrac{z-1}{2}$

(b) $x+5=2-4y$, $z=3$

Given a line with vector equation $\mathbf r=(\mathbf i+2\mathbf j+3\mathbf k)+\lambda(\mathbf i+2\mathbf j+5\mathbf k)$, $\lambda\in\mathbb R$,

(i) show that point $A(3,6,13)$ lies on the line,

(ii) find the perpendicular distance from point $B(1,7,4)$ to the line.

Given three points $A(0,2,7)$, $B(5,-3,2)$ and $C(1,1,1)$, find the position vector of the point $R$ on $AB$ such that $CR$ is perpendicular to $AB$. Hence find the perpendicular distance from $C$ to $AB$ and the position vector of the reflection of $C$ in $AB$.

Find whether the following pairs of lines are parallel, intersecting or skew. If they intersect, find the point of intersection and the acute angle between the lines.

(a) $\mathbf r=(1,-1,1)+\lambda(3,4,1)$, $\lambda\in\mathbb R$, and $\mathbf r=(1,2,3)+\mu(9,12,3)$, $\mu\in\mathbb R$.

(b) $\mathbf r=(1,0,3)+s(2,1,1)$, $s\in\mathbb R$, and $\mathbf r=(2,-1,1)+t(1,-2,0)$, $t\in\mathbb R$.

(c) $\mathbf r=4\mathbf i+8\mathbf j+3\mathbf k+\alpha(\mathbf i+2\mathbf j+\mathbf k)$, $\alpha\in\mathbb R$, and $\mathbf r=7\mathbf i+6\mathbf j+5\mathbf k+\beta(6\mathbf i+4\mathbf j+5\mathbf k)$, $\beta\in\mathbb R$.

(d) $\mathbf r=(4,2,1)+\lambda(2,1,-2)$, $\lambda\in\mathbb R$, and the z-axis.

[Hint: The z-axis has equation $\mathbf r=\gamma(0,0,1)$, $\gamma\in\mathbb R$.]

Given the two lines $l_1:4(1-x)=y=2z+4$ and $l_2:8-4x=y+3=2z$, explain why they are parallel. Find the distance between the two lines.