If a, b and c are perpendicular to b + c, c + | vedclass

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

Question

(D) Given that $a \perp (b + c)$,$b \perp (c + a)$,and $c \perp (a + b)$.This implies:$a \cdot (b + c) = 0 \implies a \cdot b + a \cdot c = 0$ .....$(

Vedclass · Accepted Answer

$10$

Vedclass · Answer

(D) Given that $a \perp (b + c)$,$b \perp (c + a)$,and $c \perp (a + b)$.This implies:$a \cdot (b + c) = 0 \implies a \cdot b + a \cdot c = 0$ .....$(i)$$b \cdot (c + a) = 0 \implies b \cdot c + b \cdot a = 0$ .....$(ii)$$c \cdot (a + b) = 0 \implies c \cdot a + c \cdot b = 0$ .....$(iii)$Adding $(i), (ii),$ and $(iii)$,we get $2(a \cdot b + b \cdot c + c \cdot a) = 0$.Now,consider the sum of the squares of the magnitudes:$|a + b|^2 + |b + c|^2 + |c + a|^2 = 6^2 + 8^2 + 10^2 = 36 + 64 + 100 = 200$.Expanding the left side:$(|a|^2 + |b|^2 + 2a \cdot b) + (|b|^2 + |c|^2 + 2b \cdot c) + (|c|^2 + |a|^2 + 2c \cdot a) = 200$$2(|a|^2 + |b|^2 + |c|^2) + 2(a \cdot b + b \cdot c + c \cdot a) = 200$.Since $2(a \cdot b + b \cdot c + c \cdot a) = 0$,we have $2(|a|^2 + |b|^2 + |c|^2) = 200$,so $|a|^2 + |b|^2 + |c|^2 = 100$.We know that $|a + b + c|^2 = |a|^2 + |b|^2 + |c|^2 + 2(a \cdot b + b \cdot c + c \cdot a)$.Substituting the values: $|a + b + c|^2 = 100 + 0 = 100$.Therefore,$|a + b + c| = \sqrt{100} = 10$.

If $a, b$ and $c$ are perpendicular to $b + c, c + a$ and $a + b$ respectively,and if $|a + b| = 6, |b + c| = 8$ and $|c + a| = 10$,then $|a + b + c| = $

Similar Questions

Let $a = 2i - j + k$,$b = i + 2j - k$,and $c = i + j - 2k$ be three vectors. $A$ vector in the plane of $b$ and $c$ whose projection on $a$ is of magnitude $\sqrt{2/3}$ is

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be vectors of equal magnitude such that the angle between $\vec{a}$ and $\vec{b}$ is $\alpha$, $\vec{b}$ and $\vec{c}$ is $\beta$, and $\vec{c}$ and $\vec{a}$ is $\gamma$. Then the minimum value of $\cos \alpha + \cos \beta + \cos \gamma$ is

If $\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}$ and $\vec{b}=\hat{i}+3 \hat{j}-5 \hat{k},$ then show that the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ are perpendicular.

The moment of the force $\overrightarrow{F}$ acting at a point $P$,about the point $C$ is:

Let $\hat{a}$ and $\hat{b}$ be two unit vectors. If the vectors $\bar{c}=\hat{a}+2 \hat{b}$ and $\bar{d}=5 \hat{a}-4 \hat{b}$ are perpendicular to each other,then the angle between $\hat{a}$ and $\hat{b}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module