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Chapter 6: Q16E (page 331)

Determine which pairs of vectors in Exercises \(15 - 18\) are orthogonal.

16. \({\bf{u}} = \left( {\begin{aligned}{12}\\3\\{ - 5}\end{aligned}} \right),\,\,{\bf{v}} = \left( {\begin{aligned}{2}\\{ - 3}\\3\end{aligned}} \right)\)

Short Answer

Expert verified

The vectors \({\bf{u}}{\rm{ and }}{\bf{v}}\) are orthogonal to each other.

Step by step solution

01

Definition of orthogonal vectors

The two vectors \({\bf{u}}{\rm{ and }}{\bf{v}}\) are Orthogonal if \({\bf{u}} \cdot {\bf{v}} = 0\).

02

Checking Orthogonality for given vectors.

The given vectors are:

\({\bf{u}} = \left( {\begin{aligned}{*{20}{r}}{12}\\3\\{ - 5}\end{aligned}} \right),\,\,{\bf{v}} = \left( {\begin{aligned}{*{20}{r}}2\\{ - 3}\\3\end{aligned}} \right)\)

On having dot products, we get:

\(\begin{aligned}{c}{\bf{u}} \cdot {\bf{v}} &= \left( {\begin{aligned}{*{20}{r}}{12}\\3\\{ - 5}\end{aligned}} \right).\left( {\begin{aligned}{*{20}{r}}2\\{ - 3}\\3\end{aligned}} \right)\\ &= \left( {12} \right)\left( 2 \right) + \left( 3 \right)\left( { - 3} \right) + \left( { - 5} \right)\left( 3 \right)\\ &= 24 - 9 - 15\\ &= 0\end{aligned}\)

Since \({\bf{u}} \cdot {\bf{v}} = 0\).

Hence, the vectors \({\bf{u}}{\rm{ and }}{\bf{v}}\) are orthogonal to each other.

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Most popular questions from this chapter

In Exercises 17 and 18, all vectors and subspaces are in \({\mathbb{R}^n}\). Mark each statement True or False. Justify each answer.

a. If \(W = {\rm{span}}\left\{ {{x_1},{x_2},{x_3}} \right\}\) with \({x_1},{x_2},{x_3}\) linearly independent,

and if \(\left\{ {{v_1},{v_2},{v_3}} \right\}\) is an orthogonal set in \(W\) , then \(\left\{ {{v_1},{v_2},{v_3}} \right\}\) is a basis for \(W\) .

b. If \(x\) is not in a subspace \(W\) , then \(x - {\rm{pro}}{{\rm{j}}_W}x\) is not zero.

c. In a \(QR\) factorization, say \(A = QR\) (when \(A\) has linearly

independent columns), the columns of \(Q\) form an

orthonormal basis for the column space of \(A\).

In Exercises 1-4, find a least-sqaures solution of \(A{\bf{x}} = {\bf{b}}\) by (a) constructing a normal equations for \({\bf{\hat x}}\) and (b) solving for \({\bf{\hat x}}\).

1. \(A = \left[ {\begin{aligned}{{}{}}{ - {\bf{1}}}&{\bf{2}}\\{\bf{2}}&{ - {\bf{3}}}\\{ - {\bf{1}}}&{\bf{3}}\end{aligned}} \right]\), \({\bf{b}} = \left[ {\begin{aligned}{{}{}}{\bf{4}}\\{\bf{1}}\\{\bf{2}}\end{aligned}} \right]\)

In Exercises 9-12, find a unit vector in the direction of the given vector.

9. \(\left( {\begin{aligned}{*{20}{c}}{ - 30}\\{40}\end{aligned}} \right)\)

Suppose \(A = QR\) is a \(QR\) factorization of an \(m \times n\) matrix

A (with linearly independent columns). Partition \(A\) as \(\left[ {\begin{aligned}{{}{}}{{A_1}}&{{A_2}}\end{aligned}} \right]\), where \({A_1}\) has \(p\) columns. Show how to obtain a \(QR\) factorization of \({A_1}\), and explain why your factorization has the appropriate properties.

Find an orthonormal basis of the subspace spanned by the vectors in Exercise 4.
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