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Chapter 11: Problem 56

Give a geometric description of the projection of \(\mathbf{u}\) onto \(\mathbf{v}\).

Short Answer

Expert verified
The projection of \(\mathbf{u}\) onto \(\mathbf{v}\), denoted as proj_\(\mathbf{v}\)\(\mathbf{u}\) or sometimes as \(\mathbf{u}_{\mathbf{v}}\), is a vector that lies along the direction of \(\mathbf{v}\) and whose length is determined by the dot product of \(\mathbf{u}\) and \(\mathbf{v}\), divided by the magnitude of \(\mathbf{v}\) squared. In other words, it is the 'shadow' that \(\mathbf{u}\) casts onto \(\mathbf{v}\), represented by the vector \(\frac{\mathbf{u}\cdot\mathbf{v}}{\|\mathbf{v}\|^2}\mathbf{v}\). The exact form of this vector will depend upon the specific components of \(\mathbf{u}\) and \(\mathbf{v}\).

Step by step solution

01

Understand vector projection

First of all, recall that the projection of vector \(\mathbf{u}\) onto vector \(\mathbf{v}\) (denoted as proj_\(\mathbf{v}\)\(\mathbf{u}\)) is given by the formula \(\frac{\mathbf{u}\cdot\mathbf{v}}{\|\mathbf{v}\|^2}\mathbf{v}\). This formula is derived from the dot product and its geometric interpretation. It essentially scales vector \(\mathbf{v}\) so that it matches the 'shadow' cast by \(\mathbf{u}\) onto \(\mathbf{v}\).
02

Apply the formula for vector projection

To find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\), substitute the appropriate components of \(\mathbf{u}\) and \(\mathbf{v}\) into the formula and calculate the resulting vector. This vector proj_\(\mathbf{v}\)\(\mathbf{u}\) is the geometric description of the projection of \(\mathbf{u}\) onto \(\mathbf{v}\).
03

Interpret the result

The resulting vector is the geometric image of \(\mathbf{u}\) that lies along the line of \(\mathbf{v}\). If \(\mathbf{u}\) and \(\mathbf{v}\) are pointing in the same general direction, the projection will be a scaled up version of \(\mathbf{v}\). If they are perpendicular, the projection will be zero, and if they are pointing in exact opposite directions, the projection shall be a scaled down version of \(\mathbf{v}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dot Product
The dot product is a fundamental concept when working with vectors and is essential for understanding vector projections. It is calculated by taking two vectors, say \(\mathbf{u}\) and \(\mathbf{v}\), and finding the sum of the products of their corresponding components. Mathematically, it is expressed as:
  • \(\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \cdots + u_nv_n\)
The dot product serves several purposes:
  • It helps to determine the angle between two vectors. The closer the dot product is to zero, the closer the vectors are to being perpendicular.
  • It is used to calculate the projection of one vector onto another.
In the context of vector projection, the dot product reflects how much of one vector extends in the direction of the other, essentially quantifying their alignment.
Geometric Interpretation
Geometrically, the projection of a vector \(\mathbf{u}\) onto another vector \(\mathbf{v}\) can be visualized as a shadow of \(\mathbf{u}\) when light is cast along a direction perpendicular to \(\mathbf{v}\). This concept emphasizes understanding not only the magnitude but also the directional component of vectors.
  • Imagine a scenario where light is shining perpendicularly to \(\mathbf{v}\). The projection of \(\mathbf{u}\) onto \(\mathbf{v}\) would be like a shadow or outline of \(\mathbf{u}\) on the vector \(\mathbf{v}\).
  • This shadow is found using the formula: \( \operatorname{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \mathbf{v} \).
The resulting vector is parallel to \(\mathbf{v}\) and shows the extent to which \(\mathbf{u}\) follows the direction of \(\mathbf{v}\). Its length is the scalar that comes out of the projection formula, indicating how much of \(\mathbf{u}\) 'fits' along \(\mathbf{v}\).
Perpendicular Vectors
Perpendicular vectors hold a unique place in vector analysis because they have a dot product of zero. This is a key indicator that the two vectors do not share any component in the direction of each other.
  • When \(\mathbf{u}\) and \(\mathbf{v}\) are perpendicular, their dot product \(\mathbf{u} \cdot \mathbf{v} = 0\).
  • This means \(\|\mathbf{v}\|\) has no influence on \(\mathbf{u}\) whatsoever, leading the projection \( \operatorname{proj}_{\mathbf{v}} \mathbf{u}\) to be the zero vector \(\mathbf{0}\).
The zero vector is a significant outcome because it portrays that vector \(\mathbf{u}\) has no component of itself directed along \(\mathbf{v}\). Understanding perpendicular vectors is crucial as they help define planes and axes in more complex geometric interpretations. Perpendicularity ensures that transformations and interpretations of vectors maintain orthogonality in various applications.

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Most popular questions from this chapter

The vertices of a triangle are given. Determine whether the triangle is an acute triangle, an obtuse triangle, or a right triangle. Explain your reasoning. $$ (-3,0,0),(0,0,0),(1,2,3) $$

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Determine which of the following are defined for nonzero vectors \(\mathbf{u}, \mathbf{v}\), and \(\mathbf{w} .\) Explain your reasoning. (a) \(\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})\) (b) \((\mathbf{u} \cdot \mathbf{v}) \mathbf{w}\) (c) \(\mathbf{u} \cdot \mathbf{v}+\mathbf{w}\) (d) \(\|\mathbf{u}\| \cdot(\mathbf{v}+\mathbf{w})\)

find the area of the triangle with the given vertices. (Hint: \(\frac{1}{2}\|\mathbf{u} \times \mathbf{v}\|\) is the area of the triangle having \(\mathbf{u}\) and \(\mathbf{v}\) as adjacent sides. $$ (0,0,0),(1,2,3),(-3,0,0) $$

Use the shell method to find the volume of the solid below the surface of revolution and above the \(x y\) -plane. The curve \(z=\sin y(0 \leq y \leq \pi)\) in the \(y z\) -plane is revolved about the \(z\) -axis.
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