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Contents Index Calc Dark Mode Prev Up Next \(\newcommand{\R}{\mathbb{R}}
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Preview Activity 9.6.1 .
We will consider what happens when we allow movement in
\(\R^3\) with the restriction that the movement must be orthogonal to
\(\vv=\langle 1,2,3 \rangle \text{.}\)
(a) Find values for
\(a_0\) and
\(b_0\) such that
\(\langle 0,a_0,b_0 \rangle\) is orthogonal to
\(\langle 1,2,3\rangle\text{.}\)
(b) Find values for
\(c_0\) and
\(d_0\) such that
\(\langle c_0,d_0,0 \rangle\) is orthogonal to
\(\langle 1,2,3\rangle\text{.}\)
(c) Find values for
\(c_1\) and
\(d_1\) such that
\(\langle c_1,d_1,1 \rangle\) is orthogonal to
\(\langle 1,2,3\rangle\text{.}\)
(d) Find two other values for each of
\(c\) and
\(d\) such that
\(\langle c,d,1 \rangle\) is orthogonal to
\(\langle 1,2,3\rangle\text{.}\)
(e)
Verify that each of the following vectors is also orthogonal to \(\langle 1,2,3 \rangle\text{.}\)
(f) Put all of the vectors you have computed into the interactive below to visually verify that each of them is orthogonal to
\(\langle 1,2,3 \rangle\text{.}\) You should put the component values of each vector into this array with each vector corresponding to a row. If you do not have eight distinct vectors from the previous parts, multiply one of your repeated vectors by
\(-1\) and enter that instead of entering a vector multiple times.
Figure 9.6.1. A plot of your vectors that should be orthogonal to \(\langle 1,2,3 \rangle\)
(g) Describe what you think the plot of the set of vectors that are orthogonal to
\(\langle 1,2,3 \rangle\) will look like.
