If P hat{i}-2 hat{j}+3 hat{k}, 2 hat{i}+3 hat | vedclass

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

Question

(D) Let the position vectors be $\vec{a} = P \hat{i} - 2 \hat{j} + 3 \hat{k}$, $\vec{b} = 2 \hat{i} + 3 \hat{j} - 4 \hat{k}$, and $\vec{c} = 4 \hat{i}

Vedclass · Accepted Answer

$\frac{1}{5 \sqrt{3}}(\hat{i}+5 \hat{j}-7 \hat{k})$

Vedclass · Answer

(D) Let the position vectors be $\vec{a} = P \hat{i} - 2 \hat{j} + 3 \hat{k}$, $\vec{b} = 2 \hat{i} + 3 \hat{j} - 4 \hat{k}$, and $\vec{c} = 4 \hat{i} + 13 \hat{j} - 18 \hat{k}$.Since the points $A$, $B$, and $C$ are collinear, the vectors $\vec{AB}$ and $\vec{BC}$ must be parallel.$\vec{AB} = \vec{b} - \vec{a} = (2-P) \hat{i} + 5 \hat{j} - 7 \hat{k}$$\vec{BC} = \vec{c} - \vec{b} = (4-2) \hat{i} + (13-3) \hat{j} + (-18+4) \hat{k} = 2 \hat{i} + 10 \hat{j} - 14 \hat{k}$Since $\vec{AB} = k \vec{BC}$, we have $(2-P) \hat{i} + 5 \hat{j} - 7 \hat{k} = k(2 \hat{i} + 10 \hat{j} - 14 \hat{k})$.Comparing the coefficients of $\hat{j}$, we get $5 = 10k \Rightarrow k = 0.5$.Comparing the coefficients of $\hat{i}$, we get $2-P = 2k = 2(0.5) = 1 \Rightarrow P = 1$.Thus, $\vec{AB} = (2-1) \hat{i} + 5 \hat{j} - 7 \hat{k} = \hat{i} + 5 \hat{j} - 7 \hat{k}$.The magnitude $|\vec{AB}| = \sqrt{1^2 + 5^2 + (-7)^2} = \sqrt{1 + 25 + 49} = \sqrt{75} = 5 \sqrt{3}$.The unit vector in the direction of $\vec{AB}$ is $\hat{u} = \frac{\vec{AB}}{|\vec{AB}|} = \frac{1}{5 \sqrt{3}} (\hat{i} + 5 \hat{j} - 7 \hat{k})$.Since $|P| = |1| = 1$, the required vector is $1 	imes \hat{u} = \frac{1}{5 \sqrt{3}} (\hat{i} + 5 \hat{j} - 7 \hat{k})$.

If $P \hat{i}-2 \hat{j}+3 \hat{k}$, $2 \hat{i}+3 \hat{j}-4 \hat{k}$, and $4 \hat{i}+13 \hat{j}-18 \hat{k}$ are the position vectors of three collinear points $A$, $B$, and $C$ respectively, then the vector in the direction of $AB$ of length $|P|$ units is

Similar Questions

If the points $P, Q$ and $R$ have the position vectors $\hat{i}-2 \hat{j}+3 \hat{k}$,$-2 \hat{i}+3 \hat{j}+2 \hat{k}$ and $-8 \hat{i}+13 \hat{j}$ respectively,then these points are

Write two different vectors having the same magnitude.

$A$ vector $\vec{a}$ has components $2p$ and $1$ with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise sense. If,with respect to the new system,$\vec{a}$ has components $p+1$ and $1$,then:

$D, E, F$ are points on the sides $BC, CA$ and $AB$ of a $\triangle ABC$ respectively,dividing them in the ratios $2:3, 1:2, 3:1$ internally. The lines $BE$ and $CF$ intersect on the line $AD$ at $P$. If $\overrightarrow{AP} = x_1 \overrightarrow{AB} + y_1 \overrightarrow{AC}$,then $x_1 + y_1 =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module