If a and b are two vectors such that | a + b | = | a |, then prov… - Vidyadip

Question

If $\overrightarrow{\text{a}}\text{ and } \overrightarrow{\text{b}}$are two vectors such that $|\overrightarrow{\text{a}} + \overrightarrow{\text{b}}| = | \overrightarrow{\text{a}}|,$ then prove that vector 2 $\overrightarrow{\text{a}} + \overrightarrow{\text{b}}$is perpendicular to vector$\overrightarrow{\text{b}}.$

✓

Answer

$\because|\overrightarrow{\text{a}} + \overrightarrow{\text{b}}| = |\overrightarrow{\text{a}}|\Rightarrow|\overrightarrow{\text{a}} +\overrightarrow{\text{b}}|^{2} =|\overrightarrow{\text{a}}|^{2}$
$\Rightarrow(\overrightarrow{\text{a}} + \overrightarrow{\text{b}}).(\overrightarrow{\text{a}} + \overrightarrow{\text{b}}) = |\overrightarrow{\text{a}}|^{2}$
$\Rightarrow\overrightarrow{\text{a}}.\overrightarrow{\text{a}} + \overrightarrow{\text{a}}.\overrightarrow{\text{b}} + \overrightarrow{\text{b}}.\overrightarrow{\text{a}}+\overrightarrow{\text{b}}.\overrightarrow{\text{b}} = |\overrightarrow{\text{a}}|^{2}$
$\Rightarrow|\overrightarrow{\text{a}}|^{2} + 2 \overrightarrow{\text{a}}.\overrightarrow{\text{b}} +\overrightarrow{\text{b}}.\overrightarrow{\text{b}} = | \overrightarrow{\text{a}}|^{2}\big[\because\overrightarrow{\text{a}}.\overrightarrow{\text{b}} = \overrightarrow{\text{b}}.\overrightarrow{\text{a}}\big]$
$\Rightarrow2\overrightarrow{\text{a}}.\overrightarrow{\text{b}} + \overrightarrow{\text{b}}.\overrightarrow{\text{b}} = 0 \Rightarrow(2\overrightarrow{\text{a}} +\overrightarrow{\text{b}}).\overrightarrow{\text{b}} = 0 $
$\Rightarrow( 2 \overrightarrow{\text{a}} + \overrightarrow{\text{b}})$ is perpendicular to $\overrightarrow{\text{b}}.$

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 "5 Marks Questions" from VECTOR ALGEBRA→VECTOR ALGEBRA — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Show that the plane whose vector equation is $\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}})=3$ contains the line whose vector equation is $\vec{\text{r}}=\hat{\text{i}}+\hat{\text{j}}+\lambda(2\hat{\text{i}}+\hat{\text{j}}+4\hat{\text{k}}).$

Give an example of a function:
Which is not one-one but onto.

Differentiate the following functions with respect to x:
$\log\sqrt{\frac{\text{x}-1}{\text{x}+1}}$

If $\big|\vec{\text{a}}+\vec{\text{b}}\big|=60,\big|\vec{\text{a}}-\vec{\text{b}}\big|=40$ and $\big|\vec{\text{b}}\big|=46,$ find $|\vec{\text{a}}|$

Evalute the following integrals:
$\int\frac{-\sin\text{x}+2\cos\text{x}}{2\sin\text{x}+\cos\text{x}}\text{dx}$

A box manufacturer makes large and small boxes from a large piece of cardboard. The large boxes require $4$ sq. metre per box while the small boxes require $3$ sq. metre per box. The manufacturer is required to make at least three large boxes and at least twice as many small boxes as large boxes. If $60$ sq. metre of cardboard is in stock, and if the profits on the large and small boxes are $Rs. 3$ and $Rs. 2$ per box, how many of each should be made in order to maximize the total profit?

Determine the points on the curve $x^2 = 4y$ which are nearest to the point $(0, 5).$

If $\text{y}=\text{x}^\text{n}\{\text{a}\cos(\log\text{x})+\text{b}\sin(\log\text{x})\},$ prove that $\text{x}^2\frac{\text{d}^2\text{y}}{\text{dx}^2}+(1-2\text{n})\frac{\text{dy}}{\text{dx}}+(1+\text{n}^2)\text{y}=0.$

If $\text{A} = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix} \text{and B = } \text{A} = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5 \end{bmatrix}$ are two square matrices, find AB and hence solve the system of linear
equations x – y = 3, 2x + 3y + 4z = 17 and y + 2z = 7.

Find the shortest distance between the following pairs of lines whose vector equation are:
$\vec{\text{r}}=\big(3\hat{\text{i}}+5\hat{\text{j}}+7\hat{\text{k}}\big)+\lambda\big(\hat{\text{i}}-2\hat{\text{j}}+7\hat{\text{k}}\big)$ and $\vec{\text{r}}=-\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}+\mu\big(7\hat{\text{i}}-6\hat{\text{j}}+\hat{\text{k}}\big)$