Textbook Question
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
-4w
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8. Vectors
Geometric Vectors
2:44 minutesProblem 33
Textbook Question
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
3v - 4w
Verified step by step guidance1
Identify the given vectors: \( \mathbf{v} = -3\mathbf{i} + 7\mathbf{j} \) and \( \mathbf{w} = -\mathbf{i} - 6\mathbf{j} \).
Multiply vector \( \mathbf{v} \) by the scalar 3: calculate \( 3\mathbf{v} = 3(-3\mathbf{i} + 7\mathbf{j}) \).
Multiply vector \( \mathbf{w} \) by the scalar 4: calculate \( 4\mathbf{w} = 4(-\mathbf{i} - 6\mathbf{j}) \).
Subtract the vector \( 4\mathbf{w} \) from \( 3\mathbf{v} \): compute \( 3\mathbf{v} - 4\mathbf{w} \) by subtracting corresponding components.
Write the resulting vector in component form \( a\mathbf{i} + b\mathbf{j} \) by combining the \( \mathbf{i} \) and \( \mathbf{j} \) components from the subtraction.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Representation in Component Form
Vectors in two dimensions can be expressed as components along the i (x-axis) and j (y-axis) unit vectors. For example, u = 2i - 5j means the vector has an x-component of 2 and a y-component of -5. This form allows for straightforward addition, subtraction, and scalar multiplication of vectors.
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Scalar Multiplication of Vectors
Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). For instance, multiplying vector v by 3 means multiplying both its i and j components by 3, resulting in a new vector scaled in magnitude but with the same direction if the scalar is positive.
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Vector Addition and Subtraction
Adding or subtracting vectors is done component-wise by adding or subtracting their corresponding i and j components. For example, to find 3v - 4w, first multiply each vector by its scalar, then subtract the resulting components to get the final vector.
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