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
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8. Vectors
Geometric Vectors
2:09 minutesProblem 26
Textbook Question
In Exercises 25–26, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.
P₁ = (-3, 0), P₂ = (-2, -2)
Verified step by step guidance1
Identify the coordinates of the initial point \(P_1\) and the terminal point \(P_2\). Here, \(P_1 = (-3, 0)\) and \(P_2 = (-2, -2)\).
Recall that the vector \(\mathbf{v}\) from \(P_1\) to \(P_2\) can be found by subtracting the coordinates of \(P_1\) from \(P_2\): \(\mathbf{v} = (x_2 - x_1, y_2 - y_1)\).
Calculate the change in the \(x\)-coordinate: \(x_2 - x_1 = -2 - (-3) = -2 + 3\).
Calculate the change in the \(y\)-coordinate: \(y_2 - y_1 = -2 - 0\).
Express the vector \(\mathbf{v}\) in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) as \(\mathbf{v} = (x_2 - x_1)\mathbf{i} + (y_2 - y_1)\mathbf{j}\).
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Representation in the Plane
A vector in the plane can be represented as a combination of unit vectors i and j, where i points in the x-direction and j points in the y-direction. Writing a vector in terms of i and j involves expressing its horizontal and vertical components as multiples of these unit vectors.
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Finding the Vector Between Two Points
The vector from an initial point P₁(x₁, y₁) to a terminal point P₂(x₂, y₂) is found by subtracting coordinates: v = (x₂ - x₁)i + (y₂ - y₁)j. This gives the displacement vector that points from P₁ to P₂.
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Unit Vectors i and j
Unit vectors i and j are standard basis vectors in two-dimensional space, where i = (1, 0) and j = (0, 1). They serve as building blocks for expressing any vector in the plane by scaling and adding these vectors according to the vector's components.
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