Find a unit vector perpendicular to each of the vectors ( a + b )… - Vidyadip

Question

Find a unit vector perpendicular to each of the vectors $\left( {\vec a + \vec b} \right)$ and $\left( {\vec a - \vec b} \right)$, where $\vec a = \hat i + \hat j + \hat k, \ \vec b = \hat i + 2\hat j + 3\hat k$.

✓

Answer

$\vec a+\vec b=(\vec i+\vec j+\vec k)+(\vec i+2\vec j+3\vec k)=2\vec i+3\vec j+4\vec k$
$\vec a-\vec b=(\vec i+\vec j+\vec k)-(\vec i+2\vec j+3 \vec k=-\vec j-2\vec k$
A vector which is perpendicular to both $(\vec a + \vec b)$ are $(\vec a - \vec b)$ is given by
$(\vec a + \vec b) \times (\vec a - \vec b) = \left| {\begin{array}{*{20}{c}} {\hat i}&{\hat j}&{\hat k} \\ 2&3&4 \\ 0&{ - 1}&{ - 2} \end{array}} \right|$
$ = - 2\hat i + 4\hat j - 2\hat k$
Let $\vec c = - 2\hat i + 4\hat j - 2\hat k$
$\left| {\vec c} \right| = \sqrt {4 + 16 + 4} $
$ = \sqrt {24} $
$ = 2\sqrt 6 $
Required unit vector is
$\frac{{\vec c}}{{\left| {\vec c} \right|}} = - \frac{1}{{\sqrt 6 }}\hat i + \frac{2}{{\sqrt 6 }}\hat j - \frac{1}{{\sqrt 6 }}\hat k$

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 "3 Marks Question" 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

Solve the following equation:
$2\tan^{-1}\left(\cos x\right)=\tan^{-1}\left(2\ \text{cosec}\ x\right)$

A vector $\vec{\text{r}}$ is inclined at equal angles to the three axes. If the magnitude of $\vec{\text{r}}$ is $2\sqrt3$, find $\vec{\text{r}}$.

Evaluate the following:
$\tan\frac{1}{2}\Big(\cos^{-1}\frac{\sqrt5}{3}\Big)$

Of the students in a college, it is known that 60% reside in a hostel and 40% do not reside in hostel. Previous year results report that 30% of students residing in hostel attain A grade and 20% of ones not residing in hostel attain A grade in their annual examination. At the end of the year, one students is chosen at random from the college and he has an A grade. What is the probability that the selected student is a hosteler?

Four cards are drawn simultaneously from a well shuffled pack of 52 playing cards. Find the probability distribution of the number of aces.

Differentiate the function $x^{x}-2^{\sin x}$ w.r.t. x.

Assume that on an average one telephone number out of 15 called between 2 P.M. and 3 P.M. on week days is busy. What is the probability that if six randomly selected telephone numbers are called, at least 3 of them will be busy?

Find fog and gof if:f(x) = x + 1, g(x) = 2x + 3

A die is rolled. If the outcome is an odd number, what is the probability that it is prime?

Give examples of two functions f: N → Z and g: Z → Z such that gof is injective but g is not injective.
(Hint: Consider f(x) = x and g(x) = |x|).