Let a, b and c be three vectors such that | a | = 3, | b | = 4, |… - Vidyadip

Question

Let $\vec a,\vec b$ and $\vec c$ be three vectors such that $\left| {\vec a} \right| = 3,\left| {\vec b} \right| = 4,\left| {\vec c} \right| = 5$ and each one of them being perpendicular to the sum of the other two, find $\left| {\vec a + \vec b + \vec c} \right|$.

✓

Answer

$\vec a.\left( {\vec b + \vec c} \right) = 0,\vec b.\left( {\vec c + \vec a} \right) = 0,\vec c.\left( {\vec a + \vec b} \right) = 0$ (given)
${\left| {\vec a + \vec b + \vec c} \right|^2} = \left( {\vec a + \vec b + \vec c} \right).\left( {\vec a + \vec b + \vec c} \right)$
$= \vec a.\vec a + \vec a\left( {\vec b + \vec c} \right) + \vec b.\vec b + \vec b\left( {\vec a + \vec c} \right) + \vec c.\vec c + \left( {\vec a + \vec b} \right)$
$= {\left| {\vec a} \right|^2} + {\left| {\vec b} \right|^2} + {\left| {\vec c} \right|^2}$
$ = 9 + 16 + 25$
= 50
$\left| {\vec a + \vec b + \vec c} \right| = \sqrt {50} $
$ = 5\sqrt 2 $

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from Vector Algebra→Vector Algebra — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman's time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman's time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftman's time.

What number of rackets and bats must be made if the factory is to work at full capacity?
If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find the maximum profit of the factory when it works at full capacity.

Solve the following differential equation:
$\text{xy}\frac{\text{dy}}{\text{dx}}=\text{x}^2-\text{y}^2$

Evaluate: $\int\limits_{0}^{\pi}\frac{4\text{x}\sin\text{x}}{1 + \cos^{2}\text{x}}\text{dx}.$

If the axes are rectangular and p is the point (2, 3, -1), find the equation of the plane throught p at right angles to OP.

Solve the following determinant equations:
$\begin{vmatrix}1&1&\text{x}\\\text{p}+1&\text{p}+1&\text{p}+\text{x}\\3&\text{x}+1&\text{x}+2\end{vmatrix}=0$

For each of the differential equation in find the particular solution satisfying the given condition:

$2\text{xy}+\text{y}^2-2\text{x}^2\frac{\text{dy}}{\text{dx}}=0;\ \text{y}=2\ \text{when x}=1$

Prove that $\frac{\text{dy}}{\text{dx}}\Big\{\frac{\text{x}}{2}\sqrt{\text{a}^2-\text{x}^2}+\frac{\text{a}^2}{2}\sin^{-1}\frac{\text{x}}{\text{a}}\Big\}=\sqrt{\text{a}^2-\text{x}^2}$

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{xy}=\text{c}^2\text{ at }\big(\text{ct},\frac{\text{c}}{\text{t}}\big)$

Write the vector equation of the following lines and hence determine the distance between them $\frac{\text{x}-1}{2}=\frac{\text{y}-2}{3}=\frac{\text{z}+4}{6}$ and $\frac{\text{x}-3}{4}=\frac{\text{y}-3}{6}=\frac{\text{z}+5}{12}$

Solve the matrix equation $\begin{bmatrix}5 & 4 \\1 & 1 \end{bmatrix}\text{X}=\begin{bmatrix}1 & -2 \\1 & 3 \end{bmatrix},$ where X is a 2 × 2 matrix.