Textbook Question
In Exercises 13–20, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.P₁ = (-1, 7), P₂ = (-7, -7)
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8. Vectors
Geometric Vectors
1:38 minutesProblem 23
Textbook Question
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
u - v
Verified step by step guidance1
Identify the given vectors: \( \mathbf{u} = 2\mathbf{i} - 5\mathbf{j} \) and \( \mathbf{v} = -3\mathbf{i} + 7\mathbf{j} \).
Recall that vector subtraction \( \mathbf{u} - \mathbf{v} \) is performed by subtracting the corresponding components of \( \mathbf{v} \) from \( \mathbf{u} \).
Subtract the \( \mathbf{i} \)-components: \( 2 - (-3) = 2 + 3 \).
Subtract the \( \mathbf{j} \)-components: \( -5 - 7 = -5 - 7 \).
Write the resulting vector as \( (2 + 3)\mathbf{i} + (-5 - 7)\mathbf{j} \), which is the vector \( \mathbf{u} - \mathbf{v} \).
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Video duration:
1mKey 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. Understanding this form allows for straightforward vector operations like addition and subtraction.
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Vector Subtraction
Vector subtraction involves subtracting corresponding components of two vectors. For vectors u and v, u - v is found by subtracting the x-components and y-components separately, resulting in a new vector. This operation is essential for finding the difference or relative position between vectors.
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Unit Vectors i and j
Unit vectors i and j represent the standard basis vectors along the x and y axes, respectively. They have a magnitude of one and direction along their respective axes. Expressing vectors in terms of i and j simplifies calculations and visualization in the plane.
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