With reference to the origin $O$, the position vectors of the points $A$ and $B$ are given as&

Question

With reference to the origin $O$, the position vectors of the points $A$ and $B$ are given as $\mathbf{a}$ and $\mathbf{b}$ respectively. It is given that the area of triangle $OAB$ is $16$ units$^{2}$ and $\left| \mathbf{b} \right|=5$ units. Find the exact value of $\left| \mathbf{a}\times \mathbf{d} \right|$, where $\mathbf{d}$ is a unit vector of $\mathbf{b}$. Give the geometrical interpretation of $\left| \mathbf{a}\times \mathbf{d} \right|$ in relation to the triangle $OAB$.[3] It is also given that $\left| \mathbf{a} \right|=8$ and $\measuredangle AOB$ is obtuse. Find the value of $\mathbf{a}\cdot \mathbf{b}$.[3] The point $C$ divides $AB$ in the ratio $\mu :1-\mu $, where $0<\mu <1$, and $\left| \mathbf{c} \right|=\frac{\sqrt{401}}{7}$ units. By considering a suitable scalar product, find the vector(s) $\overrightarrow{OC}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.[4]

Accepted Answer

(i) $|\mathbf a\times\mathbf d|=\dfrac{32}{5}$, the perpendicular height to base $OB$, or the shortest distance from $A$ to $OB$ (ii) $\mathbf a\mathbin{\cdot}\mathbf b=-24$ (iii) $\overrightarrow{OC}=\dfrac27\mathbf a+\dfrac57\mathbf b$ or $\overrightarrow{OC}=\dfrac{412}{959}\mathbf a+\dfrac{547}{959}\mathbf b$

2022 HCI P1 Q10

2022 HCI P1 Q10

Video Solution:

Solution:

Solution locked