If in the shown parallelogram `bar(AC) = -hat(i) + hat(j) - 3hat(k)` and `bar(BC) = 2 hat(i) + 5 hat(K)` then `bar(BD)` is

Options

A. `(5)/(2) hat(i) - (1)/(2) hat(j) + (13)/(2) hat(k)`
B. `5hat(i) - hat(j) + 13hat(k)`
C. `-5hat(i) + hat(j) - 13hat(k)`
D. `-(5)/(2) hat(i) + (1)/(2) hat(j) - (13)/(2)hat(k)`

Answer

B

Explanation

The shown figure is a parallelogram. Then,
`bar(AB) + bar(BC) = bar(AC)` ……(i)
And `bar(BA) + bar(AD) + bar(AD) = bar(BD)` …..(ii)
Also, `bar(AB) = -bar(BA)` and `bar(BC) = bar(AD)` ….(iii)
so, from (ii) & (iii)
`-bar(AB) + bar(BC) = bar(BD)` ....(iv)
Adding (i) & (iv)
`2bar(BC) = bar(AC) + bar(BD)`
`bar(BD) = 2bar(BC) - bar(AC)`