Textbook Question
Find the angle between each pair of vectors. Round to two decimal places as necessary.
2i + 2j, -5i - 5j
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8. Vectors
Dot Product
3:22 minutesProblem 8
Textbook Question
In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. projᵥᵥv
Verified step by step guidance1
Identify the vectors involved: here, both vectors are \( \mathbf{v} = -5\mathbf{i} + 2\mathbf{j} \) and \( \mathbf{w} = 2\mathbf{i} - 4\mathbf{j} \), but the problem asks for \( \text{proj}_{\mathbf{v}} \mathbf{v} \), which means the projection of \( \mathbf{v} \) onto itself.
Recall the formula for the projection of a vector \( \mathbf{a} \) onto a vector \( \mathbf{b} \):
\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \right) \mathbf{b} \]
where \( \mathbf{a} \cdot \mathbf{b} \) is the dot product of \( \mathbf{a} \) and \( \mathbf{b} \).
Since we are projecting \( \mathbf{v} \) onto itself, substitute \( \mathbf{a} = \mathbf{v} \) and \( \mathbf{b} = \mathbf{v} \) into the formula:
\[ \text{proj}_{\mathbf{v}} \mathbf{v} = \left( \frac{\mathbf{v} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \right) \mathbf{v} \]
Calculate the dot product \( \mathbf{v} \cdot \mathbf{v} \) by multiplying corresponding components and summing:
\[ \mathbf{v} \cdot \mathbf{v} = (-5)(-5) + (2)(2) \]
Simplify the fraction \( \frac{\mathbf{v} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \) which will equal 1, so the projection \( \text{proj}_{\mathbf{v}} \mathbf{v} \) is simply \( \mathbf{v} \) itself.
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Projection
Vector projection of one vector onto another is the component of the first vector in the direction of the second. It is found by scaling the second vector by the scalar projection, which involves the dot product divided by the magnitude squared of the second vector.
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Introduction to Vectors
Dot Product of Vectors
The dot product is an algebraic operation that takes two equal-length sequences of numbers (vectors) and returns a single number. It is calculated as the sum of the products of corresponding components and is used to find angles between vectors and projections.
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Vector Magnitude
The magnitude (or length) of a vector is the distance from the origin to the point represented by the vector in space. It is calculated using the Pythagorean theorem as the square root of the sum of the squares of its components.
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