Question

Let v1 = 2i j + k and v2 = i + j k, then the angle between v1 & v2 and a vector perpendicular to both v1 & v2 shall be :

Answer 1

Answers:

Required angle = cos⁻¹ (4 / √18)

Perpendicular vector = - j + k

Step-by-step explanation:

Given vectors are

v₁ = (2, 1, 1) and v₂ = (1, 1, 1)

So |v₁| = √(2² + 1² + 1²) = √6 and

|v₂| = √(1² + 1² + 1²) = √3

Then the angle between the two vectors v₁ and v₂ is

= cos⁻¹ {(v₁ . v₂) /(|v₁| |v₂|)}

= cos⁻¹ [{(2, 1, 1) . (1, 1, 1)} / (√6 *√3)]

= cos⁻¹ {(2 + 1 + 1) / (√6 * √3)}

= cos⁻¹ (4 / √18)

The perpendicular vector on both v₁ amd v₂ is determined by their cross product or vector product.

| i j k |

∴ v₁ × v₂ = | 2 1 1 |

| 1 1 1 |

= i (1 - 1) - j (2 - 1) + k (2 - 1)

= - j + k

This is the required perpendicular vector.

Remark:

Any vector r = xi + yj + zk can be written as r = (x, y, z)