Let y and L be as in Example 3 and Figure 3. Compute the orthogonal projection ŷ of y onto L using u = [ 2 1 ] instead of the u in Example 3. EXAMPLE 3 Let y = [ 7 6 ] and u = [ 4 2 ] . Find the orthogonal projection of y onto u . Then write y as the sum of two orthogonal vectors , one in Span { u } and one orthogonal to u . FIGURE 3 The orthogonal projection of y onto a line L through the origin.
Let y and L be as in Example 3 and Figure 3. Compute the orthogonal projection ŷ of y onto L using u = [ 2 1 ] instead of the u in Example 3. EXAMPLE 3 Let y = [ 7 6 ] and u = [ 4 2 ] . Find the orthogonal projection of y onto u . Then write y as the sum of two orthogonal vectors , one in Span { u } and one orthogonal to u . FIGURE 3 The orthogonal projection of y onto a line L through the origin.
Solution Summary: The author explains how the orthogonal projection of y onto cu is determined by the subspace L spanned by
Let y and L be as in Example 3 and Figure 3. Compute the orthogonal projection ŷ of y onto L using u =
[
2
1
]
instead of the u in Example 3.
EXAMPLE 3 Let y =
[
7
6
]
and u =
[
4
2
]
. Find the orthogonal projection of y onto u. Then write y as the sum of two orthogonal vectors, one in Span {u} and one orthogonal to u.
FIGURE 3 The orthogonal projection of y onto a line L through the origin.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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