Textbook Question
In Exercises 33–38, find projᵥᵥ v. Then decompose v into two vectors, v₁ and v₂, where v₁ is parallel to w and v₂ is orthogonal to w.
v = i + 3j, w = -2i + 5j
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8. Vectors
Dot Product
3:57 minutesProblem 39
Textbook Question
In Exercises 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector.
5u ⋅ (3v - 4w)
Verified step by step guidance1
First, express the vectors u, v, and w in component form: \( u = \langle -1, 1 \rangle \), \( v = \langle 3, -2 \rangle \), and \( w = \langle 0, -5 \rangle \).
Calculate the vector inside the parentheses: \( 3v - 4w \). Multiply each vector by the scalar and then subtract: \( 3v = 3 \times \langle 3, -2 \rangle = \langle 9, -6 \rangle \) and \( 4w = 4 \times \langle 0, -5 \rangle = \langle 0, -20 \rangle \). Then, \( 3v - 4w = \langle 9, -6 \rangle - \langle 0, -20 \rangle \).
Perform the subtraction component-wise: \( \langle 9 - 0, -6 - (-20) \rangle = \langle 9, 14 \rangle \).
Multiply the vector u by the scalar 5: \( 5u = 5 \times \langle -1, 1 \rangle = \langle -5, 5 \rangle \).
Finally, compute the dot product of \( 5u \) and \( 3v - 4w \): \( \langle -5, 5 \rangle \cdot \langle 9, 14 \rangle = (-5)(9) + (5)(14) \).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Scalar Multiplication
Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). This operation changes the magnitude of the vector without altering its direction unless the scalar is negative, which reverses the direction.
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Vector Addition and Subtraction
Vector addition and subtraction are performed component-wise. To add or subtract vectors, add or subtract their corresponding components, resulting in a new vector that combines the effects of the original vectors.
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Dot Product of Vectors
The dot product is a scalar obtained by multiplying corresponding components of two vectors and summing the results. It measures the extent to which two vectors point in the same direction and is calculated as u ⋅ v = u₁v₁ + u₂v₂ for 2D vectors.
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