In a quadrilateral ABCD, the point P divides | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let the position vectors of vertices $A, B, C, D$ be $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ respectively.Since $P$ divides $DC$ in the ratio $1:3$ i

Vedclass · Accepted Answer

-$4$

Vedclass · Answer

(D) Let the position vectors of vertices $A, B, C, D$ be $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ respectively.Since $P$ divides $DC$ in the ratio $1:3$ internally, the position vector of $P$ is $\vec{p} = \frac{3\vec{d} + 1\vec{c}}{1+3} = \frac{3\vec{d} + \vec{c}}{4}$.Since $Q$ is the mid-point of $AC$, the position vector of $Q$ is $\vec{q} = \frac{\vec{a} + \vec{c}}{2}$.Now, $\vec{PQ} = \vec{q} - \vec{p} = \frac{\vec{a} + \vec{c}}{2} - \frac{3\vec{d} + \vec{c}}{4} = \frac{2\vec{a} + 2\vec{c} - 3\vec{d} - \vec{c}}{4} = \frac{2\vec{a} + \vec{c} - 3\vec{d}}{4}$.Given expression: $\vec{AB} + \vec{AD} + \vec{BC} - 2\vec{DC} = (\vec{b} - \vec{a}) + (\vec{d} - \vec{a}) + (\vec{c} - \vec{b}) - 2(\vec{c} - \vec{d})$.Simplifying this: $\vec{b} - \vec{a} + \vec{d} - \vec{a} + \vec{c} - \vec{b} - 2\vec{c} + 2\vec{d} = -2\vec{a} - \vec{c} + 3\vec{d} = -(2\vec{a} + \vec{c} - 3\vec{d})$.Comparing with $\lambda \vec{PQ}$:$-(2\vec{a} + \vec{c} - 3\vec{d}) = \lambda \left( \frac{2\vec{a} + \vec{c} - 3\vec{d}}{4} \right)$.Thus, $\lambda = -4$.

In a quadrilateral $ABCD$, the point $P$ divides $DC$ in the ratio $1:3$ internally and $Q$ is the mid-point of $AC$. If $\vec{AB} + \vec{AD} + \vec{BC} - 2\vec{DC} = \lambda \vec{PQ}$, then the value of $\lambda$ is

Similar Questions

Answer the following as true or false.
$\vec{a}$ and $-\vec{a}$ are collinear.

If $P(2, 3, 1)$ and $Q(7, 15, 1)$,then $|\overrightarrow{PQ}| = \dots$

The position vectors of $A$ and $B$ are $(\hat{i}+\hat{j}+\hat{k})$ and $(\frac{1}{3} \hat{j}+\frac{1}{3} \hat{k})$. If $B$ divides the line segment $AC$ in the ratio $2:1$,then the position vector of $C$ is

The vector $\frac{1}{3} (2\hat{i} - 2\hat{j} + \hat{k})$ is ....

If $\vec{a}, \vec{b}, \vec{c}$ are three vectors such that $|\vec{a}|=3, |\vec{b}|=5, |\vec{c}|=7$,then $|\vec{a}-\vec{b}|^2+|\vec{b}-\vec{c}|^2+|\vec{c}-\vec{a}|^2$ does not exceed:

Vedclass Test Series

Exam Paper Generator

Online Exam Module