Question

Let a⃗ be any vector such that |a⃗| = a. The value of |a⃗ × î|² + |a⃗ × ĵ|² + |a⃗ × k̂|² is :

If a, b, c are three vectors such that a · b = a · c and a × b = a × c, a ≠ 0, then show that b = c.

For two vectors a⃗ and b⃗ Assertion (A): |a⃗ × b⃗|² + (a⃗ · b⃗)² = |a⃗|²|b⃗|² Reason (R): |a⃗ × b⃗| = (a⃗ · b⃗) tan θ, (θ ≠ π/2) Select the correct answer: (A) Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.

For two vectors a⃗ and b⃗ Assertion (A): |a⃗ × b⃗|² + (a⃗ · b⃗)² = |a⃗|²|b⃗|² Reason (R): |a⃗ × b⃗| = (a⃗ · b⃗) tan θ, (θ ≠ π/2)

If a, b and c are three unit vectors such that a + b + c = 0, prove that a×b = b×c = c×a.