In the given figure, vec.OA = vec.a and vec.OB = vec.b. If vec.AC = 3vec.AB, find vec.OC | Vector Geometry | SciPiPupil

Question: In the given figure, $\vec{OA} = \vec{a}$ and $\vec{OB} = \vec{b}$. If $\vec{AC} = 3 \vec{AB}$, find $\vec{OC}$.

Solution:

Given: $\vec{OA} = \vec{a} \; and \; \vec{OB} = \vec{b}$, $\vec{AC} = 3 \vec{AB}$

To find: value of $\vec{OC}$

In ∆ OAB,

Using ∆ law of vector addition,

$\vec{AB} = \vec{AO} + \vec{OB}$

$or, \vec{AB} = \vec{OB} - \vec{OA}$

$or, \vec{AB} = \vec{b} - \vec{a}$

In ∆ OAC,

Using ∆ law of vector addition,

$\vec{AC} = \vec{AO} + \vec{OC}$

$or, \vec{AC} = \vec{OC} - \vec{OA}$

$or, \vec{AC} = \vec{OC} - \vec{a}$

We have,

$\vec{AC} = 3 \vec{AB}$

[ Put value of $\vec{AC}$ from equation (ii) and value of $\vec{AB}$ from equation (i) ]

$or, \vec{OC} - \vec{a} = 3( \vec{b} - \vec{a})$

$or, \vec{OC} = 3\vec{b} -3vec{a} + \vec{a}$

$\therefore \vec{OC} = -2\vec{a} + 3\vec{b}$

Hence, the required value of $\vec{OC}$ is $-2\vec{a} + 3\vec{b}$.