Find the angle between the vectors a and b, where A = i + 2j - k and B

1.	Find the angle between the vectors a and b, where A = i + 2j - k and B = -i + j - 2k1. 90°2. 30°3. 60°4. 0°
Answer» Correct Answer - Option 3 : 60° Given: a = i + 2j – k and b = -i + j – 2k Concept used: Angle between two vectors: \(\cos θ = \frac{{⃗ a.⃗ b}}{{\left\| {⃗ a} \right\|\left\| {⃗ b} \right\|}}\) For two vectors \(⃗ a = \overrightarrow {{a_1}} \hat i + \overrightarrow {{a_2}} \hat j + \overrightarrow {{a_3}} \hat k\) \(⃗ b = \overrightarrow {{b_1}} \hat i + \overrightarrow {{b_2}} \hat j + \overrightarrow {{b_3}} \hat k\) \(⃗ a.⃗ b = {a_1}{b_1} + {a_2}{b_2} + {a_3}{b_3}\) Calculation: \(\vec a.\vec b \) = -1 + 2 + 2 ⇒ \(\vec a.\vec b \) = 3 ⇒ \(\vec a\) = √(1² + 2² + 1²) ⇒ \(\vec a\) = √(1 + 4 + 1) ⇒ \(\vec a\) = √6 ⇒ \(\vec b \) = √(1² + 1² + 2²) ⇒ \(\vec b \) = √1 + 1 + 4) ⇒ \(\vec b \) = √6 \(\cos θ = \frac{{⃗ a.⃗ b}}{{\left\| {⃗ a} \right\|\left\| {⃗ b} \right\|}}\) ⇒ cosθ = 3/√6 × √6 ⇒ cosθ = 3/6 ⇒cosθ = 1/2 ⇒ cosθ = 60° ∴ The angle between a and b is 60°

Find the angle between the vectors a and b, where A = i + 2j - k and B = -i + j - 2k1. 90°2. 30°3. 60°4. 0°

Answer» Correct Answer - Option 3 : 60°

Given:

a = i + 2j – k and b = -i + j – 2k

Concept used:

Angle between two vectors:

\(\cos θ = \frac{{⃗ a.⃗ b}}{{\left| {⃗ a} \right|\left| {⃗ b} \right|}}\)

For two vectors

\(⃗ a = \overrightarrow {{a_1}} \hat i + \overrightarrow {{a_2}} \hat j + \overrightarrow {{a_3}} \hat k\)

\(⃗ b = \overrightarrow {{b_1}} \hat i + \overrightarrow {{b_2}} \hat j + \overrightarrow {{b_3}} \hat k\)

\(⃗ a.⃗ b = {a_1}{b_1} + {a_2}{b_2} + {a_3}{b_3}\)

Calculation:

\(\vec a.\vec b \) = -1 + 2 + 2

⇒ \(\vec a.\vec b \) = 3

⇒ \(\vec a\) = √(1² + 2² + 1²)

⇒ \(\vec a\) = √(1 + 4 + 1)

⇒ \(\vec a\) = √6

⇒ \(\vec b \) = √(1² + 1² + 2²)

⇒ \(\vec b \) = √1 + 1 + 4)

⇒ \(\vec b \) = √6

\(\cos θ = \frac{{⃗ a.⃗ b}}{{\left| {⃗ a} \right|\left| {⃗ b} \right|}}\)

⇒ cosθ = 3/√6 × √6

⇒ cosθ = 3/6

⇒cosθ = 1/2

⇒ cosθ = 60°

∴ The angle between a and b is 60°