If M and N are the mid-points of the sides BC | vedclass

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

Question

(C) Let $\vec{AB} = \vec{a}$ and $\vec{AD} = \vec{b}$. Since $ABCD$ is a parallelogram, $\vec{BC} = \vec{AD} = \vec{b}$ and $\vec{CD} = \vec{AB} = \ve

Vedclass · Accepted Answer

$\frac{3}{2} AC$

Vedclass · Answer

(C) Let $\vec{AB} = \vec{a}$ and $\vec{AD} = \vec{b}$. Since $ABCD$ is a parallelogram, $\vec{BC} = \vec{AD} = \vec{b}$ and $\vec{CD} = \vec{AB} = \vec{a}$.Given that $M$ is the mid-point of $BC$, we have $\vec{BM} = \frac{1}{2} \vec{BC} = \frac{1}{2} \vec{b}$.Given that $N$ is the mid-point of $CD$, we have $\vec{DN} = \frac{1}{2} \vec{CD} = \frac{1}{2} \vec{a}$.In $	riangle ABM$, by triangle law of vector addition:$\vec{AM} = \vec{AB} + \vec{BM} = \vec{a} + \frac{1}{2} \vec{b}$.In $	riangle ADN$, by triangle law of vector addition:$\vec{AN} = \vec{AD} + \vec{DN} = \vec{b} + \frac{1}{2} \vec{a}$.Adding these two vectors:$\vec{AM} + \vec{AN} = (\vec{a} + \frac{1}{2} \vec{b}) + (\vec{b} + \frac{1}{2} \vec{a})$$= (1 + \frac{1}{2}) \vec{a} + (1 + \frac{1}{2}) \vec{b}$$= \frac{3}{2} \vec{a} + \frac{3}{2} \vec{b} = \frac{3}{2} (\vec{a} + \vec{b})$.Since $\vec{AC} = \vec{AB} + \vec{BC} = \vec{a} + \vec{b}$, we get:$\vec{AM} + \vec{AN} = \frac{3}{2} \vec{AC}$.

If $M$ and $N$ are the mid-points of the sides $BC$ and $CD$ respectively of a parallelogram $ABCD$, then $AM + AN$ equals

Similar Questions

If $\vec{a}, \vec{b}, \vec{c}$ are position vectors of points $A, B, C$ respectively,with $2 \vec{a}+3 \vec{b}-5 \vec{c}=\vec{0}$,then the ratio in which point $C$ divides segment $AB$ is

The angle made by the position vector of the point $(5, -4, -3)$ with the positive direction of the $X$-axis is:

If $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$,$\vec{b} = 2\hat{i} + 3\hat{j} + \hat{k}$,$\vec{c} = 3\hat{i} + \hat{j} + 2\hat{k}$ and $\alpha \vec{a} + \beta \vec{b} + \gamma \vec{c} = -3(\hat{i} - \hat{k})$. Then the triplet $(\alpha, \beta, \gamma)$ is

If $a$ and $b$ are two vectors such that $a \cdot b = 0$ and $a \times b = 0$,then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module