Relative to the origin $O$, two points $A$ and $B$ have position vectors a and b respectively.

Question

Relative to the origin $O$, two points $A$ and $B$ have position vectors a and b respectively. It is given that $\left| \mathbf{a} \right|=2,\left| \mathbf{b} \right|=1$ and $\left| 3\mathbf{a}-2\mathbf{b} \right|=\surd (37).$ By considering the scalar product $(3\mathbf{a}-2\mathbf{b})\centerdot (3\mathbf{a}-2\mathbf{b})$ show that $\mathbf{a}\centerdot \mathbf{b}=\frac{1}{4}$ and give the geometrical meaning of $|\mathbf{a}\mathrm{ }\centerdot \mathbf{b}|.$[4] Give the geometrical meaning of $\left| (\mathbf{a}-\mathbf{b})\times \mathbf{b} \right|$ and find its exact value.[3] Write down, in terms of $\mathbf{a}$ and $\mathbf{b}$, a vector equation of the line that passes through $O$ and bisects the angle $AOB$.[1]

Accepted Answer

(i) $\left| \mathbf{a}\mathbf{.b} \right|$ is the length of projection of $\mathbf{a}$ onto $\mathbf{b}$. (ii) $\vert{}(\mathbf{a}-\mathbf{b})\times\mathbf{b}\vert{}$ is $2$ times the area of $\Delta OAB$ ; $\frac{3\sqrt{7}}{4}$ (iii) $\mathbf{r}=\lambda (\mathbf{a}+2\mathbf{b}),\,\lambda \in \mathbb{R}$.

2015 DHS P1 Q7

2015 DHS P1 Q7

Video Solution:

Solution:

Solution locked