If vectors , and the angle between & is , calculate .
Table of contents
- 0. Review of College Algebra4h 45m
- 1. Measuring Angles40m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
8. Vectors
Dot Product
Problem 6
Textbook Question
In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. v ⋅ w
Verified step by step guidance1
Recall that the dot product of two vectors \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} \) and \( \mathbf{w} = w_1 \mathbf{i} + w_2 \mathbf{j} \) is given by the formula:
\[ \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 \]
Identify the components of the given vectors:
For \( \mathbf{v} = -5 \mathbf{i} + 2 \mathbf{j} \), we have \( v_1 = -5 \) and \( v_2 = 2 \).
For \( \mathbf{w} = 2 \mathbf{i} - 4 \mathbf{j} \), we have \( w_1 = 2 \) and \( w_2 = -4 \).
Substitute the components into the dot product formula:
\[ \mathbf{v} \cdot \mathbf{w} = (-5)(2) + (2)(-4) \]
Simplify the expression by performing the multiplications and then adding the results:
\[ (-5)(2) + (2)(-4) = -10 + (-8) \]
Add the two products to find the dot product value:
\[ -10 + (-8) = -18 \]
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Dot Product of Vectors
The dot product is an algebraic operation that takes two vectors and returns a scalar. It is calculated by multiplying corresponding components and summing the results, e.g., for vectors v = (v1, v2) and w = (w1, w2), v ⋅ w = v1w1 + v2w2.
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Introduction to Dot Product
Vector Components and Notation
Vectors in two dimensions are expressed in terms of unit vectors i and j, representing the x and y directions respectively. Understanding how to identify and manipulate these components is essential for performing operations like the dot product.
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i & j Notation
Geometric Interpretation of the Dot Product
The dot product relates to the angle between two vectors: v ⋅ w = |v||w|cosθ, where θ is the angle between v and w. This relationship helps in finding the angle or understanding the projection of one vector onto another.
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Introduction to Dot Product
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