In Exercises 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector.
projᵤ (v + w)

Verified step by step guidance

1. First, find the vector sum \( \mathbf{v} + \mathbf{w} \). Given \( \mathbf{v} = 3\mathbf{i} - 2\mathbf{j} \) and \( \mathbf{w} = -5\mathbf{j} \), add the corresponding components: \( \mathbf{v} + \mathbf{w} = (3\mathbf{i} - 2\mathbf{j}) + (0\mathbf{i} - 5\mathbf{j}) = 3\mathbf{i} + (-2 - 5)\mathbf{j} = 3\mathbf{i} - 7\mathbf{j} \).
2. Recall the formula for the projection of a vector \( \mathbf{a} \) onto another vector \( \mathbf{b} \): \[ \text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \right) \mathbf{b} \] Here, \( \mathbf{a} = \mathbf{v} + \mathbf{w} \) and \( \mathbf{b} = \mathbf{u} = -\mathbf{i} + \mathbf{j} \).
3. Calculate the dot product \( (\mathbf{v} + \mathbf{w}) \cdot \mathbf{u} \). Using the components: \[ (3, -7) \cdot (-1, 1) = 3 \times (-1) + (-7) \times 1 = -3 - 7 = -10 \]
4. Calculate the dot product \( \mathbf{u} \cdot \mathbf{u} \) to find the denominator: \[ (-1, 1) \cdot (-1, 1) = (-1)^2 + 1^2 = 1 + 1 = 2 \]
5. Substitute these values into the projection formula: \[ \text{proj}_{\mathbf{u}} (\mathbf{v} + \mathbf{w}) = \left( \frac{-10}{2} \right) \mathbf{u} = -5 \mathbf{u} \] This means multiply each component of \( \mathbf{u} \) by \( -5 \) to get the projection vector.

Key Concepts

Vector Addition

Vector addition involves combining two or more vectors by adding their corresponding components. For example, if u = ai + bj and v = ci + dj, then u + v = (a + c)i + (b + d)j. This operation is essential to find the resultant vector before projection.

Vector Projection

The projection of a vector a onto another vector b, denoted proj_b(a), is the vector component of a in the direction of b. It is calculated as (a · b / |b|^2) times vector b, where '·' is the dot product and |b| is the magnitude of b. This concept helps find how much of one vector lies along another.

Dot Product of Vectors

The dot product of two vectors a and b is a scalar found by multiplying corresponding components and summing the results: a · b = a₁b₁ + a₂b₂. It measures the extent to which two vectors point in the same direction and is crucial for calculating projections.