Question

Two vectors have magnitudes 3 unit and 4 unit
respectively. What should be the angle between them if
the magnitude of the resultant is (a) 1 unit, (b) 5 unit
and (c) 7 unit.
​

Answer 1

Answer:

• The Angle (θ) between Two vectors is 180 °
• The Angle (θ) between Two vectors is 90 °
• The Angle (θ) between Two vectors is 0 °

Given:

1. First vector (P) has a magnitude of 3 units.
2. Second vector (Q) has a magnitude of 4 units.

Explanation:

$\rule{300}{1.5}$

Case-1

• Resultant (R) of the vectors is 1 unit

⇒ R = √ P² + Q² + 2 P Q cos θ

Where,

• R Denotes Resultant.
• P Denotes First vector.
• Q Denotes Second vector.
• θ Denotes Angle between two vectors.

Now,

⇒ R = √ P² + Q² + 2 P Q cos θ

Substituting the values,

⇒ 1 = √ (3)² + (4)² + 2 × 3 × 4 × cos θ

Squaring on both sides.

⇒ (1)² = (3)² + (4)² + 2 × 3 × 4 × cos θ

⇒ 1 = 9 + 16 + 24 × cos θ

⇒ 1 = 25 + 24 cos θ

⇒ 1 - 25 = 24 cos θ

⇒ - 24 = 24 cos θ

⇒ cos θ = - 24 / 24

⇒ cos θ = - 1

⇒ cos θ = cos 180°  ∵ [ cos 180° = - 1 ]

⇒ θ = 180

⇒ θ = 180°

∴ The Angle (θ) between Two vectors is 180 °.

$\rule{300}{1.5}$

$\rule{300}{1.5}$

Case-2

• Resultant (R) of the vectors is 5 unit

⇒ R = √ P² + Q² + 2 P Q cos θ

Where,

• R Denotes Resultant.
• P Denotes First vector.
• Q Denotes Second vector.
• θ Denotes Angle between two vectors.

Now,

⇒ R = √ P² + Q² + 2 P Q cos θ

Substituting the values,

⇒ 5 = √ (3)² + (4)² + 2 × 3 × 4 × cos θ

Squaring on both sides.

⇒ (5)² = (3)² + (4)² + 2 × 3 × 4 × cos θ

⇒ 25 = 9 + 16 + 24 × cos θ

⇒ 25 = 25 + 24 cos θ

⇒ 25 - 25 = 24 cos θ

⇒ 0 = 24 cos θ

⇒ cos θ = 0 / 24

⇒ cos θ = 0

⇒ cos θ = cos 90°  ∵ [ cos 90° = 0 ]

⇒ θ = 90

⇒ θ = 90°

∴ The Angle (θ) between Two vectors is 90 °.

$\rule{300}{1.5}$

$\rule{300}{1.5}$

Case-3

• Resultant (R) of the vectors is 7 unit

⇒ R = √ P² + Q² + 2 P Q cos θ

Where,

• R Denotes Resultant.
• P Denotes First vector.
• Q Denotes Second vector.
• θ Denotes Angle between two vectors.

Now,

⇒ R = √ P² + Q² + 2 P Q cos θ

Substituting the values,

⇒ 7 = √ (3)² + (4)² + 2 × 3 × 4 × cos θ

Squaring on both sides.

⇒ (7)² = (3)² + (4)² + 2 × 3 × 4 × cos θ

⇒ 49 = 9 + 16 + 24 × cos θ

⇒ 49 = 25 + 24 cos θ

⇒ 49 - 25 = 24 cos θ

⇒ 24 = 24 cos θ

⇒ cos θ = 24 / 24

⇒ cos θ = 1

⇒ cos θ = cos 0°  ∵ [ cos 0° = 1 ]

⇒ θ = 0

⇒ θ = 0°

∴ The Angle (θ) between Two vectors is 0 °.

$\rule{300}{1.5}$

Answer 2

$\huge\underline\mathtt\red{Answer:-}$

•The Angle (θ) between Two vectors is 180 °

•The Angle (θ) between Two vectors is 180 °•The Angle (θ) between Two vectors is 90 °

•The Angle (θ) between Two vectors is 180 °•The Angle (θ) between Two vectors is 90 °•The Angle (θ) between Two vectors is 0 °