Question

Find out correct option if acap + 2b cap and 5a cap - 4 b cap atare perpendicular find angle between a cap and b cap.
A. 90
B. 60
C. 120
D. 180​

Answer 1

Answer-

Option B

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Given

$\hat{a} + 2 \hat{b} \: and \: 5 \hat{a} - 4 \hat{b} \\ are \: perpendicular$

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To Find

Angle between a and b

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On doing multiplication of both Vectors by scalar or dot product

We have

$(\hat{a} + 2 \hat{b}).(5 \hat{a} - 4 \hat{b}) = 0$

Because it is a property of dot product

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On solving and applying an appropriate method we get

$= > \hat{a}.( 5\hat{a} - 4 \hat{b}) + 2 \hat{b}.(5 \hat{a} - 4 \hat{b}) = 0$

On solving brackets

$= > 5( \hat{a}. \hat{a}) - 4( \hat{a}.\hat{b}) + 10( \hat{a}.\hat{b}) - 8(\hat{b}.\hat{b}) = 0 \\ = > 5 + 6(\hat{a}.\hat{b}) - 8 = 0 \\ = > 6 \cos \theta = 3 \\ = > \cos \theta = \frac{ \cancel{3}}{ \cancel{6}} \\ = > \cos \theta = \frac{1}{2} \\ = > \huge\boxed{ \theta = 60 \degree}$

Therefore

Angle between a and b will be 60°.

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Used properties of dot product in the solution

• Dot product of two unit Vectors becomes 1.
• Dot product of two perpendicular Vectors becomes 0.
• It follow distributive property.
• (Vector A)•(Vector B) = ABcosѲ.

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Answer 2

$\bold {\huge {Your ~answer :-}}$

$\bf\huge\boxed{60°}$

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EXPLANATION—

As we know from VECTORS

when two are vectors perpendicular than their dot product is zero

$(a + 2b)(5a - 4b) = 0$

5a² + 6abcosθ - 8b² = 0

(cosθ is angle between acap and bcap)

As we also know that mentioned acap and bcap, it means that they are unit vectors and have magnitude 1.

5 + 6cosθ -8 =0

Cosθ = 3/6 =1/2

θ = 60°

HENCE, the angle between the a cap and b cap is 60°

Ans-B

$\huge{\red{\ddot{\smile}}}$