With respect to an origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and

Question

With respect to an origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ and $\mathbf{b}$ are not parallel. It is given that $B$ lies on the line segment $AC$, such that $\overrightarrow{BC}=2\mathbf{b}-k\mathbf{a}$. State the value of the constant $k$. Hence, find $\overrightarrow{OC}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.[2] $\overrightarrow{ON}$ is the projection vectors of $\overrightarrow{OB}$ onto $\overrightarrow{OA}$, where $\overrightarrow{ON}=\frac{3}{4}\mathbf{a}$ and $\mathbf{a}\cdot \mathbf{b}=4$. Show that the length of $OA$ is $\frac{4\sqrt{3}}{3}$. [2] Given also that $\mathbf{b}=\left( \begin{matrix} 2 \ -2 \ 2 \ \end{matrix} \right)$, find the angle between $\overrightarrow{OA}$ and $\overrightarrow{OB}$. [3] Given that $OADC$ is a parallelogram and $E$ lies on line segment $AC$ such that $AE:AC=2:3$, find $\overrightarrow{DE}$.[3]

Accepted Answer

(i) $k=2$, $\overrightarrow{OC}=3\mathbf b-2\mathbf a$ (ii) $OA=\frac{4\sqrt3}{3}$ (iii) $60^\circ$ (iv) $\overrightarrow{DE}=\begin{pmatrix}-2\2\-2\end{pmatrix}$

Vectors - Vector Projections

Vectors - Vector Projections

Video Solution:

Solution:

Solution locked