In Exercises 5–8, letv = -5i + 2j and w = 2i - 4jFind the specified vector, scalar, or angle.projᵥᵥv

Verified step by step guidance

1. insert step 1: Understand the problem. We need to find the projection of vector \( \mathbf{v} \) onto itself, which is denoted as \( \text{proj}_{\mathbf{v}} \mathbf{v} \).
2. insert step 2: Recall the formula for the projection of a vector \( \mathbf{a} \) onto another vector \( \mathbf{b} \), which is \( \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b} \).
3. insert step 3: Substitute \( \mathbf{v} = -5\mathbf{i} + 2\mathbf{j} \) into the formula for both \( \mathbf{a} \) and \( \mathbf{b} \). This gives us \( \text{proj}_{\mathbf{v}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v} \).
4. insert step 4: Calculate the dot product \( \mathbf{v} \cdot \mathbf{v} \). This is \( (-5)(-5) + (2)(2) \).
5. insert step 5: Simplify the expression \( \frac{\mathbf{v} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v} \) to find that the projection of a vector onto itself is the vector itself, \( \mathbf{v} \).

Key Concepts

Vector Projection

Vector projection is a way to express one vector in the direction of another. The projection of vector v onto vector w, denoted as projᵥᵥv, is calculated using the formula projᵥᵥv = (v · w / w · w) * w, where '·' represents the dot product. This concept is essential for understanding how vectors relate to each other in terms of direction and magnitude.

Dot Product

The dot product is a fundamental operation in vector algebra that combines two vectors to produce a scalar. It is calculated as v · w = v₁w₁ + v₂w₂ for vectors v = (v₁, v₂) and w = (w₁, w₂). The dot product is crucial for finding the angle between vectors and is used in the projection formula to determine how much of one vector lies in the direction of another.

Unit Vector

A unit vector is a vector with a magnitude of one, which indicates direction without regard to length. To find a unit vector in the direction of a given vector v, you divide v by its magnitude, ||v||. Understanding unit vectors is important in vector projection, as they help in normalizing vectors and simplifying calculations related to direction.