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If x hat(i)+ y hat(j)+zhat(k) is a unit ...

If `x hat(i)+ y hat(j)+zhat(k)` is a unit vector and `x:y:z=sqrt(3):2:3`, then what is the value of z?

A

`(3)/(16)`

B

3

C

`3/4`

D

2

Text Solution

Verified by Experts

The correct Answer is:
C
Let `xhat(i)+yhat(i)+zhat(k)` is a unit vector.
`:.x^(2)+y^(2)+z^(2)=1`
Given `x:y:z=2k and z=3k`
`:. (sqrt(3)k)^(2)+(2k)^(2)+(3k)^(2)=1`
`rArr 3k^(2)+4k^(2)+9k^(2)=1`
`rArr k^(2)=(1)/(16)rArr k= (1)/(4)`
Hence, `z=3k=3xx(1)/(4)=(3)/(4)`
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Knowledge Check

  • If x hat(i) + y hat(j) + z hat(k) is a unit vector and x:y:z= sqrt3:2:3 then what is the value of z?

    A
    `(3)/(16)`
    B
    3
    C
    `(3)/(4)`
    D
    2

  • If hat(i), hat(j) and hat(k) are unit vectors along x,y and z-axis respectively, the angle theta between the vector hat(i) + hat(j) + hat(k)" ""and vector" hat(j) is given by

    A
    `theta = cos^(-1) ((1)/(sqrt3))`
    B
    `theta = sin^(-1) ((1)/(sqrt3))`
    C
    `theta = cos^(-1) ((sqrt3)/(2))`
    D
    `theta = sin^(-1) ((sqrt3)/(2))`

  • The vectors hat(i)- x hat(j)- y k and hat(i) + x hat(j) + y hat(k) are orthogonal to reach other, then what is the locus of the point (x,y)?

    A
    A parabola
    B
    An ellipse
    C
    A circle
    D
    A straight line

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