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Reflections 13.7 Reflect on your understanding
In addition to the reflection activities below, re-read
Section 13.2 Terminology and notation . Be sure you understand each of the new definitions introduced in this chapter, and spend some time committing them to memory.
1. Dot product versus size of angle.
Explain how the size of the angle between two vectors (acute, obtuse, or right) can be inferred from the value of the dot product of those two vectors.
2. Geometry of orthogonal projection.
State the geometric interpretation of
\(\uproj{u}{a} \) in terms of the line through the origin parallel to
\(\uvec{a} \text{.}\)
3. Parallel bases for orthogonal projections.
Explain why
\(\uproj{u}{a} = \uproj{u}{b} \) when
\(\uvec{a} \) and
\(\uvec{b} \) are parallel (nonzero) vectors.
4. Parallel and orthogonal projections.
In each case, explain geometrically (that is, without appealing to the algebraic formula for orthogonal projection).
(a) Parallel orthogonal projection.
What is the result of
\(\uproj{u}{a} \) when
\(\uvec{u} \) is parallel to
\(\uvec{a} \text{?}\)
(b) Orthogonal orthogonal projection.
What is the result of
\(\uproj{u}{a} \) when
\(\uvec{u} \) is orthogonal to
\(\uvec{a} \text{?}\)
5. Parallel and orthogonal components.
Describe the
parallel and orthogonal components of a vector
\(\uvec{u} \) relative to vector
\(\uvec{a} \text{.}\)
6. Distance to a line.
Summarize the procedure to compute the distance in
\(\R^n \) between a point and a line through the origin.
7. Normal vectors.
Describe the concept of a
normal vector for a line in
\(\R^2 \) or plane in
\(\R^3 \text{,}\) and the relationship between the algebraic equation for the line/plane and the normal vector.
8. Cross product.
Describe the concept of the
cross product of two non-parallel vectors in
\(\R^3 \text{:}\) what it represents geometrically, and how to compute it.
9. Right-hand rule.
State the
right-hand rule for cross products in
\(\R^3 \text{.}\)
