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Reflections 11.7 Reflect on your understanding
In addition to the reflection activities below, re-read
Section 11.2 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. Vectors describe displacement.
Describe how the components of a vector are calculated from an initial point and a terminal point.
2. Geometric vector addition.
Describe vector addition geometrically: how are the two vectors in a vector sum arranged, and what are the resulting initial and terminal points of the new sum vector?
3. The zero vector, geometrically and algebraically.
(a) What geometric displacement does the zero vector represent?
(b) What is the algebraic relationship of the zero vector to vector addition?
4. Geometric vector negation.
Describe vector negation geometrically: what displacement does a negative vector represent relative to the displacement represented by the original vector?
5. Geometric vector subtraction.
Describe vector subtraction geometrically: how are the two vectors in a vector difference arranged, and what are the resulting initial and terminal points of the new difference vector?
6. Geometric scalar multiplication of a vector.
(a) Describe scalar multiplication of a vector geometrically in the case of scale factor that is a positive integer as repeated addition.
(b) Describe the geometric relationship between the displacements represented by a vector and by a scalar multiple of that vector in the case of an arbitrary (possibly non-integer) positive scale factor.
(c) What effect does a
negative scale factor have?
7. Standard basis vectors.
(a)
In two-dimensional space
\(\R^2 \text{:}\)
(i) Describe the relationship between the
\(xy\) -axes and the
standard basis vectors \(\uvec{e}_1\) and
\(\uvec{e}_2\text{.}\)
(ii)
Provide a geometric interpretation of a decomposition of a two-dimensional vector \(\uvec{v} = (v_1,v_2) \) as a linear combination
\begin{equation*}
\uvec{v} = v_1 \uvec{e}_1 + v_2 \uvec{e}_2 \text{.}
\end{equation*}
(b)
In three-dimensional space
\(\R^3 \text{:}\)
(i) Describe the relationship between the
\(xyz\) -axes and the
standard basis vectors \(\uvec{e}_1\text{,}\) \(\uvec{e}_2\text{,}\) \(\uvec{e}_3\text{.}\)
(ii)
Provide a geometric interpretation of a decomposition of a three-dimensional vector \(\uvec{v} = (v_1,v_2,v_3) \) as a linear combination
\begin{equation*}
\uvec{v} = v_1 \uvec{e}_1 + v_2 \uvec{e}_2 + v_3 \uvec{e}_3 \text{.}
\end{equation*}
8. Describe the correspondence between the algebra of vectors under the operations of vector addition and scalar multiplication and the algebra of matrices under matrix addition and scalar multiplication, and how that correspondence is realized.
