2017 H2 Mathematics Paper 1 Question 6

Vectors I: Basics, Dot and Cross Products

Answers

(i)

It is a line passing through the point A{A} with direction vector a{\mathbf{a}} and parallel to the direction vector b{\mathbf{b}}

(ii)

It is a plane perpendicular to the normal vector n.{\mathbf{n}.}
d{d} represents the displacement of the plane from the origin

(iii)

r=a+(d−a⋅nb⋅n)b{\displaystyle \mathbf{r} = \mathbf{a} + \left(\frac{d - \mathbf{a}\cdot\mathbf{n}}{\mathbf{b}\cdot\mathbf{n}}\right)\mathbf{b}}
The solution is the position vector of the point of intersection between the line and the plane

Full solutions

(i)

It is a line passing through the point A{A} with direction vector a{\mathbf{a}} and parallel to the direction vector b{\mathbf{b}}

(ii)

It is a plane perpendicular to the normal vector n.{\mathbf{n}.}
d{d} represents the displacement of the plane from the origin

(iii)

(a+tb)⋅n=da⋅n+tb⋅n=d\begin{align*} (\mathbf{a}+t\mathbf{b})\cdot\mathbf{n} &= d \\ \mathbf{a}\cdot\mathbf{n} + t\mathbf{b}\cdot\mathbf{n} &= d \\ \end{align*}
tb⋅n=d−a⋅nt=d−a⋅nb⋅n\begin{align*} t\mathbf{b}\cdot\mathbf{n} &= d - \mathbf{a}\cdot\mathbf{n} \\ t &= \frac{d - \mathbf{a}\cdot\mathbf{n}}{\mathbf{b}\cdot\mathbf{n}} \\ \end{align*}
r=a+(d−a⋅nb⋅n)b  ■\mathbf{r} = \mathbf{a} + \left(\frac{d - \mathbf{a}\cdot\mathbf{n}}{\mathbf{b}\cdot\mathbf{n}}\right)\mathbf{b} \; \blacksquare
The solution is the position vector of the point of intersection between the line and the plane