Question

The angle θ between the vectors a→=5i^−j^+k^ and b→=i^+j^−k^ is:

• A. (A) cos−1⁡(13)
• B. (B) cos−1⁡(23)
• C. (C) cos−1⁡(47)
• D. (D) None of these

Answer 1

Answer: (A)

(A) cos−1⁡(13)

Answer 2

Explanation:
We have two vectors a→=5i^−j^+k^ and b→=i^+j^−k^We know that the angle θ between two vectors a→ and a→ is:θ=cos−1⁡a→⋅b→|a→|×|b| ----(1)Therefore,a→⋅b→=(5i^−j^+k^)⋅(i^+j^−k^)⇒a→⋅b→=5i^2+5i^j^−5i^k^−i^j^−j^2+j^k^+i^k^+j^k^−k^2We know that, (i^2=j^2=k^2=1) and (i^j^=j^k^=i^k^=0)∴a→⋅b→=5−1−1=3We also know that, if a→=(xi^+yj^+zk^) then,|a→|=x2+y2+z2Therefore, according to the question,⇒|a→|=52+(−1)2+12=27⇒|b→|=12+12+(−1)2=3Putting all the above values in equation (1),⇒θ=cos−1⁡327×3⇒θ=cos−1⁡13Hence, the correct option is (A).