2010 H2 Mathematics Paper 1 Question 1

Vectors I: Basics, Dot and Cross Products

Answers

(i)

p=37{p=\frac{3}{7}}

Full solutions

(i)

49p2=9p2=949p=37  ■ since p>0\begin{align*} 49 p^2 &= 9 \\ p^2 &= \frac{9}{49} \\ p &= \frac{3}{7} \; \blacksquare \quad \textrm{ since } p>0 \\ \end{align*}

(ii)

(a+b)⋅(a−b)=a⋅a+b⋅a−a⋅b−b⋅b=∣a∣2+a⋅b−a⋅b−∣b∣2=0  ■ since ∣a∣=∣b∣\begin{align*} & (\mathbf{a}+\mathbf{b}) \cdot (\mathbf{a}-\mathbf{b}) \\ &= \mathbf{a}\cdot\mathbf{a} +\mathbf{b}\cdot\mathbf{a} - \mathbf{a}\cdot\mathbf{b} - \mathbf{b}\cdot\mathbf{b} \\ &= \left|\mathbf{a}\right|^2 + \mathbf{a}\cdot\mathbf{b} - \mathbf{a}\cdot\mathbf{b} - \left|\mathbf{b}\right|^2 \\ &= 0 \; \blacksquare \quad \textrm{ since } \left|\mathbf{a}\right|=\left|\mathbf{b}\right| \\ \end{align*}