Textbook Question
In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.
v = i + j, w = i - j
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8. Vectors
Dot Product
1:54 minutesProblem 31
Textbook Question
In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.
v = 3i, w = -4j
Verified step by step guidance1
Recall that two vectors \( \mathbf{v} \) and \( \mathbf{w} \) are orthogonal if and only if their dot product is zero, i.e., \( \mathbf{v} \cdot \mathbf{w} = 0 \).
Express the vectors \( \mathbf{v} \) and \( \mathbf{w} \) in component form. Given \( \mathbf{v} = 3\mathbf{i} \), this corresponds to \( \mathbf{v} = (3, 0) \). Similarly, \( \mathbf{w} = -4\mathbf{j} \) corresponds to \( \mathbf{w} = (0, -4) \).
Use the formula for the dot product of two vectors in component form: \[ \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 \] where \( v_1, v_2 \) are components of \( \mathbf{v} \) and \( w_1, w_2 \) are components of \( \mathbf{w} \).
Substitute the components into the dot product formula: \[ (3)(0) + (0)(-4) \].
Evaluate the expression to check if the dot product equals zero. If it does, then \( \mathbf{v} \) and \( \mathbf{w} \) are orthogonal; otherwise, they are not.
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Dot Product
The dot product of two vectors is a scalar calculated by multiplying corresponding components and summing the results. For vectors v = (v1, v2) and w = (w1, w2), the dot product is v1*w1 + v2*w2. It measures how much one vector extends in the direction of another.
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Introduction to Dot Product
Orthogonality of Vectors
Two vectors are orthogonal if their dot product equals zero. This means they are perpendicular to each other in the vector space. Checking orthogonality involves computing the dot product and verifying if it is zero.
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Vector Components and Unit Vectors
Vectors can be expressed in terms of unit vectors i and j, representing the x and y directions respectively. For example, v = 3i means v has components (3, 0), and w = -4j means w has components (0, -4). Understanding this helps in calculating the dot product.
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