Aerin H.
asked • 03/21/21Use the vectors u = <2,2>, v = <-4,3>, and w = <1,-4> to find the indicated quantity.
(u · 2v)w
(u · 2v)w = ________
State whether the result is a vector or a scalar.
The result is a:
A. Vector
B. Scalar
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2 Answers By Expert Tutors
Vahram P. answered • 03/22/21
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New to Wyzant
B.S. in math from UCLA & several years of tutoring experience
The product of a scalar and vector returns a vector so 2v is a vector. The dot product of two vectors u · 2v is a scalar since the dot product is defined as the sum of the product of the corresponding entries between the vectors. So u · 2v is a scalar. Then (u · 2v)w is, once again, the product of a scalar and a vector so it returns a vector.
Answer A: Vector.
Yefim S. answered • 03/21/21
Tutor
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Math Tutor with Experience
(u · 2v)w = (<2,2> - 2<-4,3>)·<1, -4> = <10, - 4>·<1, - 4> = 10 + 16 = 26
Answer B. Scalar
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Vahram P.
2v is scalar multiplication of a vector, so we simply scale both entries by 2. That is, 2v = < 2*(-4), 2*3> = <-8,6>. Dot product (·) is the sum of the products of the corresponding entries of the vectors, so it's always a scalar. That is, u ·2v = <2,2> · <-8,6> = 2*(-8) + 2*6 = -16 + 12 = -4. Then (u · 2v)w is also scalar multiplication of a vector, so we again scale both entries by the scalar (u · 2v) = -4. That is, (u · 2v)w = -4<1,-4> = <-4*1, -4*(-4)> = <-4,16> So the result is the vector <-4,16>.03/22/21