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Linear Algebra : Matrices

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Example Questions

Example Question #843 : Linear Algebra

$\textbf{u},\textbf{v} , \textbf{w}, \textbf{x} \in \mathbb{R}^{3}$

Which of the following applies to $(\textbf{u} \times \textbf{v}) \times( \textbf{w} \times \textbf{x})$ , where " $\cdot$ " and " $\times$ " refer to the dot product and the cross product of two vectors?

Possible Answers:

$(\textbf{u} \times \textbf{v}) \times( \textbf{w} \times \textbf{x})$ is an undefined expression.

$(\textbf{u} \times \textbf{v}) \times( \textbf{w} \times \textbf{x}) \in M_{2 \times 2}$

$(\textbf{u} \times \textbf{v}) \times( \textbf{w} \times \textbf{x}) \in \mathbb{R}^{3}$

$(\textbf{u} \times \textbf{v}) \times( \textbf{w} \times \textbf{x}) \in \mathbb{R}$

Correct answer:

$(\textbf{u} \times \textbf{v}) \times( \textbf{w} \times \textbf{x}) \in \mathbb{R}^{3}$

Explanation:

The cross product of two vectors in $\mathbb{R}^{3}$ is also a vector in $\mathbb{R}^{3}$ . It follows that $\textbf{u} \times \textbf{v} \in \mathbb{R}^{3}$ and $\textbf{w} \times \textbf{x} \in \mathbb{R}^{3}$ ; it further follows that $(\textbf{u} \times \textbf{v}) \times( \textbf{w} \times \textbf{x}) \in \mathbb{R}^{3}$ .

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Example Question #34 : Vector Vector Product

$\textbf{u},\textbf{v} \in \mathbb{R}^{3}$ ; $\textbf{w}, \textbf{x} \in \mathbb{R}^{4}$

Which of the following applies to $(\textbf{u} \times \textbf{v}) + ( \textbf{w} \times \textbf{x})$ , where " $\times$ " refers to the cross product of two vectors, and "" refers to either scalar or vector addition, as applicable?

Possible Answers:

$(\textbf{u} \times \textbf{v}) + ( \textbf{w} \times \textbf{x}) \in \mathbb{R}$

$(\textbf{u} \times \textbf{v}) + ( \textbf{w} \times \textbf{x}) \in \mathbb{R}^{3}$

$(\textbf{u} \times \textbf{v}) + ( \textbf{w} \times \textbf{x}) \in M_{2 \times 3}$

$(\textbf{u} \times \textbf{v}) + ( \textbf{w} \times \textbf{x})$ is an undefined expression.

$(\textbf{u} \times \textbf{v}) + ( \textbf{w} \times \textbf{x}) \in \mathbb{R}^{2}$

Correct answer:

$(\textbf{u} \times \textbf{v}) + ( \textbf{w} \times \textbf{x})$ is an undefined expression.

Explanation:

The cross product of two vectors is defined only if both vectors are in $\mathbb{R}^{3}$ . $\textbf{w}$ and $\textbf{x}$ are vectors in $\mathbb{R}^{4}$ , so $\textbf{w} \times \textbf{x}$ is undefined; consequently, so is $(\textbf{u} \times \textbf{v}) + ( \textbf{w} \times \textbf{x})$ .

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Example Question #842 : Linear Algebra

$\textbf{u},\textbf{v} \in \mathbb{R}^{3}$ ; $\textbf{w}, \textbf{x} \in \mathbb{R}^{4}$

Which of the following applies to $(\textbf{u} \cdot \textbf{v}) + ( \textbf{w} \cdot \textbf{x})$ , where " $\cdot$ " refers to the dot product of two vectors, and "" refers to either scalar or vector addition, as applicable?

Possible Answers:

$(\textbf{u} \cdot \textbf{v}) + ( \textbf{w} \cdot \textbf{x}) \in \mathbb{R}^{3}$

$(\textbf{u} \cdot \textbf{v}) + ( \textbf{w} \cdot \textbf{x}) \in \mathbb{R}^{4}$

$(\textbf{u} \cdot \textbf{v}) + ( \textbf{w} \cdot \textbf{x})$ is an undefined expression.

$(\textbf{u} \cdot \textbf{v}) + ( \textbf{w} \cdot \textbf{x}) \in \mathbb{R}$

$(\textbf{u} \cdot \textbf{v}) + ( \textbf{w} \cdot \textbf{x}) \in M_{3 \times 4 }$

Correct answer:

$(\textbf{u} \cdot \textbf{v}) + ( \textbf{w} \cdot \textbf{x}) \in \mathbb{R}$

Explanation:

The dot product of two vectors in the same vector space is a scalar quantity. $\textbf{u},\textbf{v} \in \mathbb{R}^{3}$ , so $\textbf{u}$ and $\textbf{v}$ are in the same vector space; their dot product is defined, and $\textbf{u} \cdot \textbf{v} \in \mathbb{R}$ . For similar reasons, $\textbf{w} \cdot \textbf{x} \in \mathbb{R}$ . Therefore, their sum is defined, and $(\textbf{u} \cdot \textbf{v}) + ( \textbf{w} \cdot \textbf{x}) \in \mathbb{R}$ .

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Example Question #35 : Vector Vector Product

$\textbf{u},\textbf{v} \in \mathbb{R}^{3}$ .

Which of the following applies to $\textbf{u} \times \textbf{v} + \textbf{u} \cdot \textbf{v}$ , where " $\cdot$ " and " $\times$ " refer to the dot product and the cross product of two vectors, and "" refers to either scalar or vector addition, as applicable?

Possible Answers:

$\textbf{u} \times \textbf{v} + \textbf{u} \cdot \textbf{v} \in M_{1 \times 3 }$

$\textbf{u} \times \textbf{v} + \textbf{u} \cdot \textbf{v} \in \mathbb{R} ^{3}$

$\textbf{u} \times \textbf{v} + \textbf{u} \cdot \textbf{v}$ is an undefined expression.

$\textbf{u} \times \textbf{v} + \textbf{u} \cdot \textbf{v} \in M_{3 \times 1 }$

$\textbf{u} \times \textbf{v} + \textbf{u} \cdot \textbf{v} \in \mathbb{R}$

Correct answer:

$\textbf{u} \times \textbf{v} + \textbf{u} \cdot \textbf{v}$ is an undefined expression.

Explanation:

The cross product $\textbf{u} \times \textbf{v}$ of two vectors in $\mathbb{R}^{3}$ is also a vector in $\mathbb{R}^{3}$ . The dot product $\textbf{u} \cdot \textbf{v}$ of two such vectors is a scalar. Since a vector and a scalar cannot be added, $\textbf{u} \times \textbf{v} + \textbf{u} \cdot \textbf{v}$ is an undefined expression.

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Example Question #32 : Vector Vector Product

$\textbf{u} \cdot \textbf{v} = 20$

True or false: It follows that $\textbf{v} \cdot \textbf{u} = 20$ .

Possible Answers:

False

True

Correct answer:

True

Explanation:

One property of vector dot products is commutativity - that is,

$\textbf{v}\cdot \textbf{u} = \textbf{u} \cdot \textbf{v}$ .

Therefore, if $\textbf{u} \cdot \textbf{v} = 20$ , then $\textbf{v} \cdot \textbf{u} = 20$ .

The statement is true,

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Example Question #33 : Vector Vector Product

$\textbf{u} \times \textbf{v}= (1, 2, 3)$

True or false: It follows that $\textbf{v}\times \textbf{u} = (1, 2, 3)$ .

Possible Answers:

False

True

Correct answer:

False

Explanation:

One property of vector cross products is anticommutativity - that is,

$\textbf{v}\times \textbf{u} = -(\textbf{u} \times \textbf{v})$ .

If $\textbf{u} \times \textbf{v}= (1, 2, 3)$ , it follows that

$\textbf{v}\times \textbf{u} = -(1,2,3) = (-1. -2. -3)$ .

The statement is false.

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Example Question #38 : Vector Vector Product

$\textbf{u},\textbf{v} , \textbf{w}, \textbf{x} \in \mathbb{R}^{3}$

Which of the following applies to $(\textbf{u} \times \textbf{v}) \cdot ( \textbf{w} \times \textbf{x})$ , where " $\cdot$ " and " $\times$ " refer to the dot product and the cross product of two vectors?

Possible Answers:

$(\textbf{u} \times \textbf{v}) \cdot ( \textbf{w} \times \textbf{x}) \in M_{2 \times 2}$

$(\textbf{u} \times \textbf{v}) \cdot ( \textbf{w} \times \textbf{x}) \in \mathbb{R}$

$(\textbf{u} \cdot \textbf{v}) \times( \textbf{w} \times \textbf{x})$ is an undefined expression.

$(\textbf{u} \times \textbf{v}) \cdot ( \textbf{w} \times \textbf{x}) \in \mathbb{R}^{3}$

Correct answer:

$(\textbf{u} \times \textbf{v}) \cdot ( \textbf{w} \times \textbf{x}) \in \mathbb{R}$

Explanation:

The cross product of two vectors in $\mathbb{R}^{3}$ is also a vector in $\mathbb{R}^{3}$ . It follows that $\textbf{u} \times \textbf{v} \in \mathbb{R}^{3}$ and $\textbf{w} \times \textbf{x} \in \mathbb{R}^{3}$ . The dot product of two vectors in the same vector space, is a scalar, so $(\textbf{u} \times \textbf{v}) \cdot ( \textbf{w} \times \textbf{x})$ , the dot product of two vectors in $\mathbb{R}^{3}$ , is a scalar in $\mathbb{R}$ .

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Example Question #33 : Vector Vector Product

$\textbf{u} = (1,x, -3 )$

$\textbf{v } = (4, 3, y)$

Find and so that $\textbf{u} \times \textbf{v} =(0 , -15, 15 )$ .

Possible Answers:

x=- 9

y= 1

x= 3

y= -3

There does not exist any such and

x= 9

y= -1

x= -3

y= 3

Correct answer:

x= -3

y= 3

Explanation:

The cross product of two vectors in $\mathbb{R}^{3}$ can be set up and calculated as if it were a determinant of a matrix with its top row comprising $\textbf{i} , \textbf{j}, \textbf{k}$ , the unit vectors of $\mathbb{R}^{3}$ , and the other two comprising the elements of $\textbf{u}$ and $\textbf{v}$ :

$\textbf{u} \times \textbf{v} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \\ 1 & x& -3 \\ 4 & 3 &y \end{vmatrix}$

Calculate as you would a determinant, adding the upper-left to lower-right products and subtracting upper-right to lower-left products:

Cross product

The cross-product is equal to

$(xy)\textbf{ i } -12\textbf{j} + 3\textbf{k } - (4x)\textbf{k} - (-9)\textbf{i} -y\textbf{j}$

$= (xy+9)\textbf{ i } + (-y -12)\textbf{j} +( -4x+ 3)\textbf{k }$

= (xy+9 , -y-12, -4x+3 )

We want this vector to be equal to (0 , -15, 15 ) , so the following must hold:

-4x+ 3 = 15

x= -3

-y - 12 = -15

y = 3

Examining the third equation, xy+9 = 0 , we find that this is consistent with the other equations, since x= -3 and y = 3 make this true. Therefore, x= -3 and y = 3 are the values sought.

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Example Question #41 : Vector Vector Product

$\textbf{u},\textbf{v} \in \mathbb{R}^{3}$ .

Which of the following holds for $\textbf{u} \times \textbf{v}+ \textbf{v} \times \textbf{u}$ ?

Note: $\textbf{0}$ (boldface) refers to the vector (0,0,0) ;

Possible Answers:

$\textbf{u} \times \textbf{v}+ \textbf{v} \times \textbf{u} \in \mathbb{R}^{3}$ , but the result need not be $\textbf{0}$ .

$\textbf{u} \times \textbf{v}+ \textbf{v} \times \textbf{u} \in \mathbb{R}$ , but the result need not be 0.

$\textbf{u} \times \textbf{v}+ \textbf{v} \times \textbf{u}$ is an undefined expression.

$\textbf{u} \times \textbf{v}+ \textbf{v} \times \textbf{u} = 0$ , where 0 refers to the real zero.

$\textbf{u} \times \textbf{v}+ \textbf{v} \times \textbf{u} = \textbf{0}$ , where $\textbf{0}$ refers to the vector (0,0,0) .

Correct answer:

$\textbf{u} \times \textbf{v}+ \textbf{v} \times \textbf{u} = \textbf{0}$ , where $\textbf{0}$ refers to the vector (0,0,0) .

Explanation:

One property of vector cross products is anticommutativity - that is,

$\textbf{v}\times \textbf{u} = -(\textbf{u} \times \textbf{v})$ .

It therefore follows that for all $\textbf{u},\textbf{v} \in \mathbb{R}^{3}$ ,

$\textbf{u} \times \textbf{v}+ \textbf{v} \times \textbf{u} = \textbf{0}$ , the zero vector (0.0.0) .

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Example Question #853 : Linear Algebra

$\textbf{u} = (4, -1,4 )$

$\textbf{v} = (2, -1, a)$

A parallelogram has these two vectors as sides. Find so that the parallelogram is a rhombus.

Possible Answers:

$a = \pm 2\sqrt{7}$

$a = \pm 6$

$a = \pm \frac{9}{4}$

$a = \pm 28$

$a = \pm \frac{3}{2}$

Correct answer:

$a = \pm 2\sqrt{7}$

Explanation:

For the parallelogram formed by $\textbf{u}$ and $\textbf{v}$ to be a rhombus, the vectors must be of equal length, or norm - $\left \| \textbf{u} \right \|= \left \| \textbf{v} \right \|$ . The norm of a vector is equal to the square root of the sum of the squares of its entries; to set the norms equal, it suffices to set the squares of the norms equal, and to solve for :

$\left \| \textbf{u} \right \|^{2}= \left \| \textbf{v} \right \|^{2}$

$4^{2}+ (-1)^{2}+ 4^{2} = 2^{2}+ (-1)^{2}+a^{2}$

$16+1+16 = 4+1 +a^{2}$

$33 = 5 +a^{2}$

$a^{2} = 28$

$a = \pm 2\sqrt{7}$

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Linear Algebra : Matrices

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #843 : Linear Algebra

Example Question #34 : Vector Vector Product

Example Question #842 : Linear Algebra

Example Question #35 : Vector Vector Product

Example Question #32 : Vector Vector Product

Example Question #33 : Vector Vector Product

Example Question #38 : Vector Vector Product

Example Question #33 : Vector Vector Product

Example Question #41 : Vector Vector Product

Example Question #853 : Linear Algebra

All Linear Algebra Resources

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