Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Two Vectors Have Magnitudes 2 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. - Physics

Two vectors have magnitudes 2 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.

#### Solution

Let the two vectors be  $\vec{a}$  and  $\vec{b}$

Now,

$\left| \vec{a} \right| = 3 \text { and } \left| \vec{b} \right| = 4$

(a) If the resultant vector is 1 unit, then

$\sqrt{\vec{a}^2 + \vec{b}^2 + 2 . \vec{a} . \vec{b} \cos \theta} = 1$

$\Rightarrow \sqrt{3^2 + 4^2 + 2 . 3 . 4 \cos \theta} = 1$

Squaring both sides, we get:

$25 + 24 \cos \theta = 1$

$\Rightarrow 24 \cos \theta = - 24$

$\Rightarrow \cos \theta = - 1$

$\Rightarrow \theta = 180^\circ$

Hence, the angle between them is 180°.
(b) If the resultant vector is 5 units, then

$\sqrt{\vec{a}^2 + \vec{b}^2 + 2 . \vec{a} . \vec{b} \cos \theta} = 5$

$\Rightarrow \sqrt{3^2 + 4^2 + 2 . 3 . 4 \cos \theta} = 5$

Squaring both sides, we get:

25 + 24 cos θ = 25
⇒ 24 cos θ = 0
⇒ cos θ = 90°

Hence, the angle between them is 90°.
(c) If the resultant vector is 7 units, then

$\sqrt{\vec{a}^2 + \vec{b}^2 + 2 . \vec{a} . \vec{b} \cos \theta} = 1$

$\Rightarrow \sqrt{3^2 + 4^2 + 2 . 3 . 4 \cos \theta} = 7$
Squaring both sides, we get:

25 + 24 cos θ = 49,
⇒ 24 cos θ = 24
⇒ cos θ = 1
⇒ θ = cos−1 1 = 0°

Hence, the angle between them is 0°.

Concept: What is Physics?
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#### APPEARS IN

HC Verma Class 11, Class 12 Concepts of Physics Vol. 1
Chapter 2 Physics and Mathematics
Exercise | Q 6 | Page 29
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