If a, b, c are three non-zero,non-coplanar ve | vedclass

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

Question

(B) The given vectors are constructed using the Gram-Schmidt orthogonalization process. For the set $\{a, b_1, c_2\}$: $1$. $a \cdot b_1 = a \cdot (b

Vedclass · Accepted Answer

$\{a, b_1, c_2\}$

Vedclass · Answer

(B) The given vectors are constructed using the Gram-Schmidt orthogonalization process. For the set $\{a, b_1, c_2\}$: $1$. $a \cdot b_1 = a \cdot (b - \frac{b \cdot a}{|a|^2} a) = a \cdot b - \frac{b \cdot a}{|a|^2} (a \cdot a) = a \cdot b - b \cdot a = 0$. Thus,$a \perp b_1$. $2$. $a \cdot c_2 = a \cdot (c - \frac{c \cdot a}{|a|^2} a - \frac{c \cdot b_1}{|b_1|^2} b_1) = a \cdot c - \frac{c \cdot a}{|a|^2} (a \cdot a) - \frac{c \cdot b_1}{|b_1|^2} (a \cdot b_1) = a \cdot c - c \cdot a - 0 = 0$. Thus,$a \perp c_2$. $3$. $b_1 \cdot c_2 = b_1 \cdot (c - \frac{c \cdot a}{|a|^2} a - \frac{c \cdot b_1}{|b_1|^2} b_1) = b_1 \cdot c - \frac{c \cdot a}{|a|^2} (b_1 \cdot a) - \frac{c \cdot b_1}{|b_1|^2} (b_1 \cdot b_1) = b_1 \cdot c - 0 - c \cdot b_1 = 0$. Thus,$b_1 \perp c_2$. Since all pairs are orthogonal,the set $\{a, b_1, c_2\}$ is a set of mutually orthogonal vectors. Therefore,option $(b)$ is the correct answer.

Similar Questions

The angle between the vectors $2 \hat{k} - 3 \hat{j}$ and $\hat{i} - 2 \hat{k}$ is

If $\theta$ is the angle between two unit vectors $a$ and $b$,then $\sin(\theta/2) = $ ......

$ABCD$ is a parallelogram. The position vectors of $A$ and $C$ are respectively $3\hat{i} + 3\hat{j} + 5\hat{k}$ and $\hat{i} - 5\hat{j} - 5\hat{k}$. If $M$ is the midpoint of the diagonal $DB$,then the magnitude of the projection of $\vec{OM}$ on $\vec{OC}$,where $O$ is the origin,is

If $\bar{a}$ and $\bar{b}$ are not perpendicular to each other,$\bar{r} \times \bar{a} = \bar{b} \times \bar{a}$ and $\bar{r} \cdot \bar{c} = 0$,then $\bar{r} =$

If $\vec{a}$,$\vec{b}$,and $\vec{a}-\vec{b}$ are unit vectors and the angle between the two vectors $\vec{a}$ and $\vec{b}$ is $\theta$,then $\theta = $ . . . . . . .

Vedclass Test Series

Exam Paper Generator

Online Exam Module