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Linear Algebra : Vector-Vector Product

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Example Question #1 : Vector Vector Product

Compute $A\cdot B$ , where

$A=\begin{bmatrix} 2\\ 3\\ 4\\ \end{bmatrix}$

$B=\begin{bmatrix} 1 & 3 & 5 \end{bmatrix}$

Possible Answers:

$A\cdot B =\begin{bmatrix} 2 & 9& 10\\ 3& 6& 15 \\ 4& 20& 12 \end{bmatrix}$

$A\cdot B =\begin{bmatrix} 2 & 6& 10\\ 3& 9& 15 \\ 4& 12& 20 \end{bmatrix}$

Not possible

Correct answer:

$A\cdot B =\begin{bmatrix} 2 & 6& 10\\ 3& 9& 15 \\ 4& 12& 20 \end{bmatrix}$

Explanation:

Before we compute the product of , and , we need to check if it is possible to take the product. We will check the dimensions. is $3\times1$ , and is $1\times3$ , so the dimensions of the resulting matrix will be $3\times3$ . Now let's compute it.

$A\cdot B=\begin{bmatrix} 2\cdot1 & 2\cdot3 & 2\cdot 5\\ 3 \cdot 1& 3\cdot3 & 3 \cdot 5 \\ 4 \cdot 1 & 4\cdot3 & 4\cdot 5 \end{bmatrix} =\begin{bmatrix} 2 & 6& 10\\ 3& 9& 15 \\ 4& 12& 20 \end{bmatrix}$

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Example Question #1 : Vector Vector Product

Find the vector-vector product of the following vectors.

$A=[1 \ 2]$

$B=\begin{bmatrix} 3\\ 5 \end{bmatrix}$

Possible Answers:

It's not possible to multiply these vectors

Correct answer:

Explanation:

$A\cdot B=1\cdot3+2\cdot5=3+10=13$

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Example Question #1 : Vector Vector Product

Calculate $\bf{A} \times \bf{B}$ , given

$\textbf{A}= 4\hat{i}-2\hat{j}$

$\textbf{B}= 1\hat{i}+3\hat{j}$

Possible Answers:

$\textbf{A} \times \textbf{B} = 14 \hat{j}$

$\textbf{A} \times \textbf{B} = 14 \hat{k}$

$\textbf{A} \times \textbf{B} = 14 \hat{i}$

$\textbf{A} \times \textbf{B} = 14$

Correct answer:

$\textbf{A} \times \textbf{B} = 14 \hat{k}$

Explanation:

By definition,

$\textbf{A} \times \textbf{B} = \begin{vmatrix} \hat{i}& \hat{j}&\hat{k} \\ 4& -2& 0\\ 1& 3& 0 \end{vmatrix} =\hat{i}(0)-\hat{j}(0)+\hat{k}[4(3)-1(-2)] = 14 \hat{k}$ .

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Example Question #4 : Vector Vector Product

What is the physical significance of the resultant vector $\bf{C}$ , if $\bf{C}= \bf{A} \times \bf{B}$ ?

Possible Answers:

$\bf{C}$ lies in the same plane that contains both $\bf{A}$ and $\bf{B}$ .

$\bf{C}$ is orthogonal to both $\bf{A}$ and $\bf{B}$ .

$\bf{C}$ is the projection of $\bf{A}$ onto $\bf{B}$ .

$\bf{C}$ is a scalar.

Correct answer:

$\bf{C}$ is orthogonal to both $\bf{A}$ and $\bf{B}$ .

Explanation:

By definition, the resultant cross product vector (in this case, $\bf{C}$ ) is orthogonal to the original vectors that were crossed (in this case, $\bf{A}$ and $\bf{B}$ ). In , this means that $\bf{C}$ is a vector that is normal to the plane containing $\bf{A}$ and $\bf{B}$ .

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Example Question #5 : Vector Vector Product

$\begin{align*}&\text{Find the product }A\times B \\&\text{Where }A=\begin{bmatrix}-18&12&-11&4&-19&13\end{bmatrix}\text{ and }B=\begin{bmatrix}-6\\-2\\-17\\-20\\-8\\10\end{bmatrix}\end{align*}$

Possible Answers:

473

$\text{The multiplication cannot be performed.}$

519

556

Correct answer:

473

Explanation:

$\begin{align*}&\text{In order to multiply two vectors, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }1\times n\text{ and }n\times 1\\&\text{Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since A has dimensions: }1\times6\\&\text{and B has dimensions: }6\times1\\&\text{The two vectors can be multiplied:}\\&\begin{bmatrix}-18&12&-11&4&-19&13\end{bmatrix}\times\begin{bmatrix}-6\\-2\\-17\\-20\\-8\\10\end{bmatrix}\\&(-18)(-6)+(12)(-2)+(-11)(-17)+(4)(-20)+(-19)(-8)+(13)(10)\\&473\end{align*}$

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Example Question #6 : Vector Vector Product

$\begin{align*}&\text{Find the vector product }\\&\begin{bmatrix}-7&-6\end{bmatrix}\times\begin{bmatrix}11\\-9\end{bmatrix}\end{align*}$

Possible Answers:

$\text{The multiplication cannot be performed.}$

Correct answer:

Explanation:

$\begin{align*}&\text{In order to multiply two vectors, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }1\times n\text{ and }n\times 1\\&\text{Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since our vectors have dimensions: }1\times2\text{ and }2\times1\\&\text{The two can be multiplied to find a product:}\\&\begin{bmatrix}-7&-6\end{bmatrix}\times\begin{bmatrix}11\\-9\end{bmatrix}\\&(-7)(11)+(-6)(-9)\\&-23\end{align*}$

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Example Question #1 : Vector Vector Product

$\begin{align*}&\text{Find the product }A\times B \\&\text{Where }A=\begin{bmatrix}18&-15&10&-15&-14&11\end{bmatrix}\text{ and }B=\begin{bmatrix}6\\-5\\-13\\9\\10\\-8\end{bmatrix}\end{align*}$

Possible Answers:

$\text{The multiplication cannot be performed.}$

Correct answer:

Explanation:

$\begin{align*}&\text{In order to multiply two vectors, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }1\times n\text{ and }n\times 1\\&\text{Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since A has dimensions: }1\times6\\&\text{and B has dimensions: }6\times1\\&\text{The two vectors can be multiplied:}\\&\begin{bmatrix}18&-15&10&-15&-14&11\end{bmatrix}\times\begin{bmatrix}6\\-5\\-13\\9\\10\\-8\end{bmatrix}\\&(18)(6)+(-15)(-5)+(10)(-13)+(-15)(9)+(-14)(10)+(11)(-8)\\&-310\end{align*}$

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Example Question #8 : Vector Vector Product

$\begin{align*}&\text{Find the value of }C\text{ if }C=A\times B \\&\text{Where }A=\begin{bmatrix}16&-1&6&-20\end{bmatrix}\text{ and }B=\begin{bmatrix}10\\19\\-19\\1\end{bmatrix}\end{align*}$

Possible Answers:

$\text{The multiplication cannot be performed.}$

Correct answer:

Explanation:

$\begin{align*}&\text{In order to multiply two vectors, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }1\times n\text{ and }n\times 1\\&\text{Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since A has dimensions: }1\times4\\&\text{and B has dimensions: }4\times1\\&\text{It is possible to find a value for }C:\\&C=\begin{bmatrix}16&-1&6&-20\end{bmatrix}\times\begin{bmatrix}10\\19\\-19\\1\end{bmatrix}\\&C=(16)(10)+(-1)(19)+(6)(-19)+(-20)(1)\\&C=7\end{align*}$

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Example Question #9 : Vector Vector Product

$\begin{align*}&\text{Find the product }A\times B \\&\text{Where }A=\begin{bmatrix}13&8\end{bmatrix}\text{ and }B=\begin{bmatrix}-12\\7\end{bmatrix}\end{align*}$

Possible Answers:

$\text{The multiplication cannot be performed.}$

Correct answer:

Explanation:

$\begin{align*}&\text{In order to multiply two vectors, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }1\times n\text{ and }n\times 1\\&\text{Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since A has dimensions: }1\times2\\&\text{and B has dimensions: }2\times1\\&\text{The two vectors can be multiplied:}\\&\begin{bmatrix}13&8\end{bmatrix}\times\begin{bmatrix}-12\\7\end{bmatrix}\\&(13)(-12)+(8)(7)\\&-100\end{align*}$

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Example Question #10 : Vector Vector Product

$\begin{align*}&\text{Find the vector product }\\&\begin{bmatrix}-11&19&13&15&-18&6\end{bmatrix}\times\begin{bmatrix}3\\-14\\9\\-15\\-16\\10\end{bmatrix}\end{align*}$

Possible Answers:

$\text{The multiplication cannot be performed.}$

Correct answer:

Explanation:

$\begin{align*}&\text{In order to multiply two vectors, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }1\times n\text{ and }n\times 1\\&\text{Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since our vectors have dimensions: }1\times6\text{ and }6\times1\\&\text{The two can be multiplied to find a product:}\\&\begin{bmatrix}-11&19&13&15&-18&6\end{bmatrix}\times\begin{bmatrix}3\\-14\\9\\-15\\-16\\10\end{bmatrix}\\&(-11)(3)+(19)(-14)+(13)(9)+(15)(-15)+(-18)(-16)+(6)(10)\\&-59\end{align*}$

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Linear Algebra : Vector-Vector Product

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #1 : Vector Vector Product

Example Question #1 : Vector Vector Product

Example Question #1 : Vector Vector Product

Example Question #4 : Vector Vector Product

Example Question #5 : Vector Vector Product

Example Question #6 : Vector Vector Product

Example Question #1 : Vector Vector Product

Example Question #8 : Vector Vector Product

Example Question #9 : Vector Vector Product

Example Question #10 : Vector Vector Product

All Linear Algebra Resources

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