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.
Answers
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:
- First vector (P) has a magnitude of 3 units.
- Second vector (Q) has a magnitude of 4 units.
Explanation:
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 °.
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 °.
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 °.
•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 °