Problem 77 - Precalculus

Question

Suppose that you are given a vector \(\mathbf{u} .\) For what vectors \(\mathbf{v}\) does proja \(\mathbf{v}=\mathbf{0} ?\)

Step-by-Step Solution

Answer

Vectors \( \mathbf{v} \) that are perpendicular to \( \mathbf{a} \) result in zero projection.

Step 1: Understand the Concept of Projection

Projection of a vector \( \mathbf{v} \) onto \( \mathbf{a} \), denoted as \( \text{proj}_{\mathbf{a}} \mathbf{v} \), is a vector that points in the direction of \( \mathbf{a} \) and has a length that is equal to the component of \( \mathbf{v} \) in the direction of \( \mathbf{a} \). The formula is given by \( \text{proj}_{\mathbf{a}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{a}}{\mathbf{a} \cdot \mathbf{a}} \mathbf{a} \).

Step 2: Determine the Condition for Zero Projection

For \( \text{proj}_{\mathbf{a}} \mathbf{v} = \mathbf{0} \), the scalar multiplier \( \frac{\mathbf{v} \cdot \mathbf{a}}{\mathbf{a} \cdot \mathbf{a}} \) must be zero. This requires that the dot product \( \mathbf{v} \cdot \mathbf{a} = 0 \).

Step 3: Understanding the Dot Product Equals Zero Condition

The dot product of two vectors is zero if and only if they are perpendicular to each other. Therefore, for \( \text{proj}_{\mathbf{a}} \mathbf{v} = \mathbf{0} \), \( \mathbf{v} \) must be perpendicular to \( \mathbf{a} \).

Step 4: Conclusion About the Vectors \( \mathbf{v} \)

Vectors \( \mathbf{v} \) that result in \( \text{proj}_{\mathbf{a}} \mathbf{v} = \mathbf{0} \) are all vectors that are orthogonal or perpendicular to the vector \( \mathbf{a} \).

Key Concepts

Dot Product

In the world of vectors, the dot product is a key operation that combines two vectors into a single scalar value. This operation gives us a measure of how much two vectors are aligned with each other. It essentially takes the length of one vector in the direction of the other. The formula for calculating the dot product \( \mathbf{u} \cdot \mathbf{v} \) is

• \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \ldots + u_nv_n \)

where \( u_1, u_2, \ldots, u_n \) and \( v_1, v_2, \ldots, v_n \) are the components of the vectors \( \mathbf{u} \) and \( \mathbf{v} \). A noteworthy point is that the dot product is zero when two vectors are perpendicular to each other.
This property of the dot product is crucial in understanding orthogonal vectors, which we will explore in the next section.

Orthogonal Vectors

Orthogonal vectors are an interesting and important concept in vector algebra as well as in geometry. Two vectors are said to be orthogonal if they are perpendicular to each other. Mathematically, this is expressed as the dot product of the two vectors being zero. Thus, if we have two vectors \( \mathbf{u} \) and \( \mathbf{v} \), they are orthogonal when

• \( \mathbf{u} \cdot \mathbf{v} = 0 \)

This orthogonality or perpendicularity implies that there is no component of one vector along the direction of the other, which leads us directly to the concept of zero projection, which we will discuss next.

Zero Projection

The concept of zero projection is closely linked to orthogonal vectors. When we talk about projecting one vector onto another and getting a result of zero, it indicates that the two vectors are orthogonal. For any given vector \( \mathbf{v} \), the projection of \( \mathbf{v} \) onto another vector \( \mathbf{a} \) is given by

• \( \text{proj}_{\mathbf{a}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{a}}{\mathbf{a} \cdot \mathbf{a}} \mathbf{a} \)

If this projection results in a zero vector, then the dot product \( \mathbf{v} \cdot \mathbf{a} \) must be zero, reaffirming that the vectors \( \mathbf{v} \) and \( \mathbf{a} \) are orthogonal.
This means that when a vector has zero projection onto another, it has no component in the direction of the second vector. This highlights the geometric interpretation of vector projection and orthogonality.