Let vec{a}, vec{b}, and vec{c} be three non-z | vedclass

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

Question

(D) Given that $\vec{a} + 2\vec{b}$ is collinear with $\vec{c}$ and $\vec{b} + 3\vec{c}$ is collinear with $\vec{a}$.This implies $\vec{a} + 2\vec{b}

Vedclass · Accepted Answer

$\vec{0}$

Vedclass · Answer

(D) Given that $\vec{a} + 2\vec{b}$ is collinear with $\vec{c}$ and $\vec{b} + 3\vec{c}$ is collinear with $\vec{a}$.This implies $\vec{a} + 2\vec{b} = k_1 \vec{c}$ and $\vec{b} + 3\vec{c} = k_2 \vec{a}$ for some non-zero scalars $k_1$ and $k_2$.From the first equation,$\vec{a} = k_1 \vec{c} - 2\vec{b}$.Substituting this into the second equation: $\vec{b} + 3\vec{c} = k_2(k_1 \vec{c} - 2\vec{b})$.Rearranging the terms: $\vec{b} + 3\vec{c} = k_1 k_2 \vec{c} - 2k_2 \vec{b}$.$(1 + 2k_2) \vec{b} + (3 - k_1 k_2) \vec{c} = \vec{0}$.Since $\vec{b}$ and $\vec{c}$ are not collinear,their coefficients must be zero:$1 + 2k_2 = 0 \Rightarrow k_2 = -\frac{1}{2}$.$3 - k_1 k_2 = 0 \Rightarrow 3 - k_1(-\frac{1}{2}) = 0 \Rightarrow 3 + \frac{k_1}{2} = 0 \Rightarrow k_1 = -6$.Substituting $k_1 = -6$ into the first equation: $\vec{a} + 2\vec{b} = -6\vec{c}$.Therefore,$\vec{a} + 2\vec{b} + 6\vec{c} = \vec{0}$.

Let $\vec{a}, \vec{b},$ and $\vec{c}$ be three non-zero vectors such that no two of them are collinear. If the vector $\vec{a} + 2\vec{b}$ is collinear with $\vec{c}$ and $\vec{b} + 3\vec{c}$ is collinear with $\vec{a}$,then $\vec{a} + 2\vec{b} + 6\vec{c} = \dots$

Similar Questions

$A$ unit vector which is in the opposite direction of $(2, -2, 1)$ is $........$

If the vector $19 \hat{i}+22 \hat{j}+5 \hat{k}$ bisects an angle between the vectors $a$ and $6 \hat{i}+8 \hat{j}$,then the unit vector in the direction of $a$ is

In a parallelogram $ABCD$,if $\vec{AC}$ and $\vec{BD}$ are the diagonals,then which of the following is equal to $\vec{AC} + \vec{BD}$?

If $\vec{a} = -4\hat{i} + 2\hat{j} - 5\hat{k}$ and $\vec{b} = 12\hat{i} - 6\hat{j} + 15\hat{k}$,then the vectors $\vec{a}$ and $\vec{b}$ are $.......$

The perimeter of the triangle whose vertices have the position vectors $\hat{i}+\hat{j}+\hat{k}$,$5\hat{i}+3\hat{j}-3\hat{k}$,and $2\hat{i}+5\hat{j}+9\hat{k}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module