Explanation Given: Let ‖ v ‖ = 2 , ‖ w ‖ = 3 , and the angle between v and w is 120 ° . Formula used: Distributive law: ( u + v ) ⋅ w = u ⋅ w + v ⋅ w Scalar multiple law: u ⋅ ( a v ) = a ( u ⋅ v ) Commutative law: u ⋅ v = v ⋅ u The dot product of v and w is v ⋅ w = ‖ v ‖ ‖ w ‖ cos θ . Calculation: Let ‖ v ‖ = 2 , ‖ w ‖ = 3 , and the angle between v and w is 120 ° . Part a: We have to find v ⋅ w . Use the formula: v ⋅ w = ‖ v ‖ ‖ w ‖ cos θ v ⋅ w = ‖ v ‖ ‖ w ‖ cos θ It is given that, ‖ v ‖ = 2 , ‖ w ‖ = 3 , and the angle between v and w is 120 ° . v ⋅ w = ( 2 ) ( 3 ) cos ( 120 ° ) = 6 ( − 1 2 ) = − 3 Hence, the value of v ⋅ w is − 3 . Part b: We have to find ‖ 2 v + w ‖ . First, take ‖ 2 v + w ‖ 2 . We can rewrite it as: ‖ 2 v + w ‖ 2 = ( 2 v + w ) ⋅ ( 2 v + w ) Use the distributive law: ( v + w ) ⋅ u = v ⋅ u + w ⋅ u . ‖ 2 v + w ‖ 2 = ( 2 v + w ) ⋅ ( 2 v + w ) = ( 2 v + w ) ⋅ 2 v + ( 2 v + w ) ⋅ w = 2 v ⋅ 2 v + w ⋅ 2 v + 2 v ⋅ w + w ⋅ w Use the scalar multiple law: u ⋅ ( a v ) = a ( u ⋅ v ) ‖ 2 v + w ‖ 2 = 4 ( v ⋅ v ) + 2 ( w ⋅ v ) + 2 ( v ⋅ w ) + w ⋅ w Use the commutative law: u ⋅ v = v ⋅ u ‖ 2 v + w ‖ 2 = 4 ( v ⋅ v ) + 2 ( v ⋅ w ) + 2 ( v ⋅ w ) + w ⋅ w Use the law: v ⋅ v = ‖ v ‖ 2 ‖ 2 v + w ‖ 2 = 4 ‖ v ‖ 2 + 4 ( v ⋅ w ) + ‖ w ‖ 2 It is given that ‖ v ‖ = 2 , ‖ w ‖ = 3 and v ⋅ w = − 3 . Simplify the above function. ‖ 2 v + w ‖ 2 = 4 ‖ v ‖ 2 + 4 ( v ⋅ w ) + ‖ w ‖ 2 = 4 ( 2 ) 2 + 4 ( − 3 ) + ( 3 ) 2 = 16 − 12 + 9 = 13 Take square root on both sides...

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Author: Rogawski

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Chapter 13.3, Problem 48E

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a. Find $v \cdot w$ .

b. Find $‖ 2 v + w ‖$ .

c. Find $‖ 2 v - 3 w ‖$ .

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Chapter 13.3, Problem 47E

Chapter 13.3, Problem 49E

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CALCULUS W/SAPLING ACCESS >IC<

Ch. 13.1 - Prob. 1PQ Ch. 13.1 - Prob. 2PQ Ch. 13.1 - Prob. 3PQ Ch. 13.1 - Prob. 4PQ Ch. 13.1 - Prob. 5PQ Ch. 13.1 - Prob. 6PQ Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E

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a. Find v ⋅ w . b. Find ‖ 2 v + w ‖ . c. Find ‖ 2 v − 3 w ‖ .

Solution Summary: The author explains the formula used to determine the value of v w.

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a. Find $v \cdot w$ .

b. Find $‖ 2 v + w ‖$ .

c. Find $‖ 2 v - 3 w ‖$ .

Want to see the full answer?

Chapter 13 Solutions

a. Find v ⋅ w . b. Find ‖ 2 v + w ‖ . c. Find ‖ 2 v − 3 w ‖ .

Solution Summary: The author explains the formula used to determine the value of v w.

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a. Find v⋅w. b. Find ‖2v+w‖. c. Find ‖2v−3w‖.

Want to see the full answer?

Chapter 13 Solutions

a. Find $v \cdot w$ .

b. Find $‖ 2 v + w ‖$ .

c. Find $‖ 2 v - 3 w ‖$ .