Question

Write each of the following expressions in unit vector notation of the form , x i + y j + z k , where x, y, and z are just numbers.

A: (i x j) x (i + k)

B: [i x (j x i)] x k

C: [(i + j) x (j+k)] x ( i x k)

D: (i + j) x [(j + k) x (i x k)]

Answer 1

A.

(i x j) x (i + k)

= k x (i + k) ; as i x j = k

= k x i + k x k

= j + 0 ; as k x k = 0

= j

B.

[i x (j x i)] x k

= [i x (-k)] x k ; as j x i = -k

= - ( i x k) x k

= - (-j) x k ; as i x k = -j

= j x k

= i

C.

[(i + j) x (j+k)] x ( i x k)

= [ i x j + j x j + i x k + j x k] x (-j) ; as i x k = -j

= [ k + 0 + (-j) + i] x (-j) ; using i x j = k, j x j = 0 , i x k =-j ,& j x k = i

= -k x j + j x j - i x j

= -(-i) + 0 - k

= i -k

D.

(i + j) x [(j + k) x (i x k)]

= (i + j) x [(j + k) x (-j)]

=(i + j) x [ - j x j - k x j]

= (i + j) x ( 0 + i)

= i x i + j x i

= 0 + (-k)

= -k