Home
Class 12
MATHS
If a unit vector vec a makes angles pi/...

If a unit vector ` vec a` makes angles `pi/3` with ` hat i ,pi/4` with ` hat j` and an acute angle `theta` with ` hat k` , then find the value of `theta` .

A

`(5pi)/(6)`

B

`(pi)/(4)`

C

`(5pi)/(12)`

D

`(2pi)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
D
Given unit vector a makes an angle `(pi)/(3) " with " hat(i) , (pi)/(4) " with " hat(j) " and " 0 in (0, pi) " with " hat(k)`
Now we know that `cos^(2) alpha+ cos^(2) beta+ cos^(2) gamma = 1` where `alpha , beta , gamma` are angles made by the vectors with respectively `hat(i) , hat(j) " and " hat(k)`
`:. cos^(2) ((pi)/(3)) + cos^(2) ((pi)/(4)) + cos^(2) 0=1`
`rArr (1)/(4) + (1)/(2) + cos^(2) 0=1 rArr cos^(2) 0= (1)/(4) rArr cos 0 = +- (1)/(2)`
`rArr cos 0 = cos .((pi)/(3) " or " cos ((2pi)/(3)) rArr 0= (pi)/(3) " or " (2pi)/(3)`
So , 0 is `(2pi)/(3),` according to options,
Promotional Banner

Similar Questions

Explore conceptually related problems

If a unit vector vec a makes angles (pi)/(3) with hat i , (pi)/(4) with hat j and an acute angle theta with hat k , then find theta and hence, the components of vec a .

If vec a.hat i= vec a.( hat i+ hat j)= vec a.( hat i+ hat j+ hat k) , then find the unit vector vec a

Vector 1/3(2i-2j+k) is (A) a unit vector (B) makes an angle pi//3 with vector (2 hat i-4 hat j+3 hat k) (C) parallel to vector (- hat i+ hat j-1/2 hat k) (D) perpendicular to vector 3 hat i+2 hat j-2 hat k

Thescalar product of the vector hat i + hat j + hat k with a unit vector along the sum of vectors 2 hat i + 4 hat j - 5 hat k and lambda hat i + 2 hat j + 3 hat k is equal to one. Find the value of lambda .

If vectors vec a= hat i+2 hat j- hat k , vec b=2 hat i- hat j+ hat ka n d vec c="lambda" hat"i"+ hat"j"+2 hat"k" are coplanar, then find the value of (lambda-4)dot

Find the unit vector along vec a - vec b where veca = hat i + 3 hat j - hat k and vec b = 3 hat i + 2 hat j + hat k .

If hat a , hat b ,a n d hat c are three unit vectors, such that hat a+ hat b+ hat c is also a unit vector and theta_1,theta_2a n dtheta_3 are angles between the vectors hat a , hat b ; hat b , hat ca n d hat c , hat a respectively, then among theta_1,theta_2,a n dtheta_3dot a. all are acute angles b. all are right angles c. at least one is obtuse angle d. none of these

For given vectors vec a = 2 hat i - hat j + 2 hat k and vec b = - hat i + hat j - hat k , find the unit vector in the direction of the vector vec a + vec b .

Find the angle between the vectors hat i - 2 hat j + 3 hat k and 3 hat i - 2 hat j + hat k .

If in parallelogram ABCD, diagonal vectors are vec A C=2 hat i+3 hat j+4 hat k and vec B D=-6 hat i+7 hat j-2 hat k , then find the adjacent side vectors vec A B and vec A D