Textbook Question
Write each vector in the form a i + b j.
〈2, 0〉
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8. Vectors
Geometric Vectors
Problem 53
Textbook Question
Find the dot product for each pair of vectors.
4i, 5i - 9j
Verified step by step guidance1
Identify the given vectors. The first vector is \(\vec{A} = 4\mathbf{i}\) and the second vector is \(\vec{B} = 5\mathbf{i} - 9\mathbf{j}\).
Recall the formula for the dot product of two vectors \(\vec{A} = a_1\mathbf{i} + a_2\mathbf{j}\) and \(\vec{B} = b_1\mathbf{i} + b_2\mathbf{j}\):
\(\vec{A} \cdot \vec{B} = a_1 b_1 + a_2 b_2\).
Extract the components of each vector: For \(\vec{A}\), \(a_1 = 4\) and \(a_2 = 0\) (since there is no \(\mathbf{j}\) component). For \(\vec{B}\), \(b_1 = 5\) and \(b_2 = -9\).
Substitute the components into the dot product formula:
\(\vec{A} \cdot \vec{B} = (4)(5) + (0)(-9)\).
Simplify the expression by multiplying and adding the terms to find the dot product.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Components and Notation
Vectors are often expressed in terms of unit vectors i, j, and k, representing the x, y, and z directions respectively. Understanding how to identify and separate these components is essential for performing operations like the dot product.
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i & j Notation
Dot Product Definition
The dot product of two vectors is a scalar obtained by multiplying corresponding components and summing the results. For vectors in two dimensions, it is calculated as (x1 * x2) + (y1 * y2), which measures the extent to which the vectors point in the same direction.
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Properties and Applications of the Dot Product
The dot product is commutative and relates to the angle between vectors through the formula A · B = |A||B|cosθ. It is used to find projections, angles, and to determine orthogonality between vectors.
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