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Linear Algebra : Vector-Vector Product

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Example Question #131 : Matrices

$\begin{align*}&\text{Find the product }A\times B \\&\text{Where }A=\begin{bmatrix}13&-7&-14\end{bmatrix}\text{ and }B=\begin{bmatrix}-11\\12\\-4\end{bmatrix}\end{align*}$

Possible Answers:

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

Correct answer:

Explanation:

$\begin{align*}&\text{In order to multiply two vectors, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }1\times n\text{ and }n\times 1\\&\text{Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since A has dimensions: }1\times3\\&\text{and B has dimensions: }3\times1\\&\text{The two vectors can be multiplied:}\\&\begin{bmatrix}13&-7&-14\end{bmatrix}\times\begin{bmatrix}-11\\12\\-4\end{bmatrix}\\&(13)(-11)+(-7)(12)+(-14)(-4)\\&-171\end{align*}$

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Example Question #132 : Matrices

$\begin{align*}&\text{Find the value of }C\text{ if }C=A\times B \\&\text{Where }A=\begin{bmatrix}-5&5&-16&-19&8\end{bmatrix}\text{ and }B=\begin{bmatrix}18\\13\\-11\\-11\\-12\end{bmatrix}\end{align*}$

Possible Answers:

169

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

264

303

Correct answer:

264

Explanation:

$\begin{align*}&\text{In order to multiply two vectors, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }1\times n\text{ and }n\times 1\\&\text{Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since A has dimensions: }1\times5\\&\text{and B has dimensions: }5\times1\\&\text{It is possible to find a value for }C:\\&C=\begin{bmatrix}-5&5&-16&-19&8\end{bmatrix}\times\begin{bmatrix}18\\13\\-11\\-11\\-12\end{bmatrix}\\&C=(-5)(18)+(5)(13)+(-16)(-11)+(-19)(-11)+(8)(-12)\\&C=264\end{align*}$

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Example Question #11 : Vector Vector Product

$\begin{align*}&\text{Find the vector product }\\&\begin{bmatrix}-18&-4&-20\end{bmatrix}\times\begin{bmatrix}-9\\-11\\-11\end{bmatrix}\end{align*}$

Possible Answers:

426

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

428

522

Correct answer:

426

Explanation:

$\begin{align*}&\text{In order to multiply two vectors, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }1\times n\text{ and }n\times 1\\&\text{Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since our vectors have dimensions: }1\times3\text{ and }3\times1\\&\text{The two can be multiplied to find a product:}\\&\begin{bmatrix}-18&-4&-20\end{bmatrix}\times\begin{bmatrix}-9\\-11\\-11\end{bmatrix}\\&(-18)(-9)+(-4)(-11)+(-20)(-11)\\&426\end{align*}$

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Example Question #12 : Vector Vector Product

$\begin{align*}&\text{Find the vector product }\\&\begin{bmatrix}-8&-19&5&-9&2\end{bmatrix}\times\begin{bmatrix}-5\\-14\\0\\10\\12\end{bmatrix}\end{align*}$

Possible Answers:

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

311

192

240

Correct answer:

240

Explanation:

$\begin{align*}&\text{In order to multiply two vectors, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }1\times n\text{ and }n\times 1\\&\text{Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since our vectors have dimensions: }1\times5\text{ and }5\times1\\&\text{The two can be multiplied to find a product:}\\&\begin{bmatrix}-8&-19&5&-9&2\end{bmatrix}\times\begin{bmatrix}-5\\-14\\0\\10\\12\end{bmatrix}\\&(-8)(-5)+(-19)(-14)+(5)(0)+(-9)(10)+(2)(12)\\&240\end{align*}$

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Example Question #13 : Vector Vector Product

$\begin{align*}&\text{Find the vector product }\\&\begin{bmatrix}1&-7&8&9&-9\end{bmatrix}\times\begin{bmatrix}-20\\-8\\-7\\-2\\-20\end{bmatrix}\end{align*}$

Possible Answers:

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

142

165

Correct answer:

142

Explanation:

$\begin{align*}&\text{In order to multiply two vectors, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }1\times n\text{ and }n\times 1\\&\text{Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since our vectors have dimensions: }1\times5\text{ and }5\times1\\&\text{The two can be multiplied to find a product:}\\&\begin{bmatrix}1&-7&8&9&-9\end{bmatrix}\times\begin{bmatrix}-20\\-8\\-7\\-2\\-20\end{bmatrix}\\&(1)(-20)+(-7)(-8)+(8)(-7)+(9)(-2)+(-9)(-20)\\&142\end{align*}$

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Example Question #14 : Vector Vector Product

$\begin{align*}&\text{Find the vector product }\\&\begin{bmatrix}-3&-11&-9&-8&3\end{bmatrix}\times\begin{bmatrix}8\\-3\\4\\-20\\8\end{bmatrix}\end{align*}$

Possible Answers:

157

193

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

Correct answer:

157

Explanation:

$\begin{align*}&\text{In order to multiply two vectors, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }1\times n\text{ and }n\times 1\\&\text{Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since our vectors have dimensions: }1\times5\text{ and }5\times1\\&\text{The two can be multiplied to find a product:}\\&\begin{bmatrix}-3&-11&-9&-8&3\end{bmatrix}\times\begin{bmatrix}8\\-3\\4\\-20\\8\end{bmatrix}\\&(-3)(8)+(-11)(-3)+(-9)(4)+(-8)(-20)+(3)(8)\\&157\end{align*}$

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Example Question #831 : Linear Algebra

Let $\bf{a}$ and $\bf{b}$ be vectors defined by

$\bf{a} = [ 9, 0, 9, 5], \bf{b} = [4, 3, 0, 9]$ .

Find the dot product $\bf{a}\cdot\bf{b}$ .

Possible Answers:

$\bf{a}\cdot\bf{b} = \begin{bmatrix}9 & 0 & 9 & 5 \\ 4 & 3 & 0 & 9 \end{bmatrix}$

$\bf{a}\cdot\bf{b} = 45$

The dimensions do not match and the dot product does not exist.

$\bf{a}\cdot\bf{b} = 81$

$\bf{a}\cdot\bf{b} = 108$

Correct answer:

$\bf{a}\cdot\bf{b} = 81$

Explanation:

Vectors $\bf{a}$ and $\bf{b}$ are both of length 4. The dimensions match and the dot product exists.

$\bf{a}\cdot\bf{b} = [ 9, 0, 9, 5]\cdot [4, 3, 0, 9] = (9)(4)+(0)(3)+(9)(0)+(5)(9)=81$

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Example Question #16 : Vector Vector Product

Let $\bf{a}$ and $\bf{b}$ be vectors defined by

$\bf{a} = [8, -1, 4], \bf{b} = [7, 4, 3]$ .

Find the cross product $\bf{a}\times\bf{b}$ .

Possible Answers:

$\bf{a}\times\bf{b} &= -20i-4j+39k$

$\bf{a}\times\bf{b} &= -19i-4j+16k$

The cross product does not exist.

$\bf{a}\times\bf{b} &= -20i+4j+32k$

$\bf{a}\times\bf{b} &= -19i+4j+39k$

Correct answer:

$\bf{a}\times\bf{b} &= -19i+4j+39k$

Explanation:

We can find the cross product by calculating the determinant of the following matrix

$\bf{a}\times\bf{b} = \det \begin{bmatrix} i& j & k\\ 8 & -1 & 4\\ 7&4&3 \end{bmatrix}$

$\begin{align*} \bf{a}\times\bf{b} &= i ((-1)(3)-(4)(4))-j((8)(3)-(4)(7))+k((8)(4)-(-1)(7)) \\ \bf{a}\times\bf{b} &= -19i+4j+39k \end{align*}$

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Example Question #17 : Vector Vector Product

Let $\bf{u}$ and $\bf{v}$ be vectors defined by

$\bf{u} = [4,-3, 5], \bf{v}=[8, -2, 4]$ .

Find the cross product $\bf{u}\times\bf{v}$ .

Possible Answers:

$\bf{u}\times \bf{v} &= -2i + 24j +16k$

$\bf{u}\times \bf{v} &= -12i + 24j -16k$

$\bf{u}\times \bf{v} &= -2i - 24j +24k$

The cross product does not exist.

$\bf{u}\times \bf{v} &= -16i + 2j +8k$

Correct answer:

$\bf{u}\times \bf{v} &= -2i + 24j +16k$

Explanation:

We find the cross product by finding the determinant of the following matrix

$\bf{u}\times \bf{v}=\det \begin{bmatrix} i & j & k\\ 4 & -3 & 5\\ 8 & -2 & 4 \end{bmatrix}$

$\begin{align*} \bf{u}\times \bf{v} &= i ((-3)(4)-(5)(-2)) - j((4)(4)-(5)(8)) + k((4)(-2)-(-3)(8)) \\ \bf{u}\times \bf{v} &= -2i + 24j +16k \end{align*}$

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Example Question #11 : Vector Vector Product

$\textbf{u} = (1, x, x^{2}, x ^{3}, x^{4}, x^{5})$

$\textbf{v} = (y^{5}, y^{4}, y^{3}, y^{2}, y, 1)$

The expression $\textbf{u} \cdot \textbf{v}$ yields a polynomial of what degree?

Possible Answers:

None of the other choices gives a correct response.

Correct answer:

Explanation:

The dot product $\textbf{u} \cdot \textbf{v}$ is the sum of the products of entries in corresponding positions, so

$\textbf{u} \cdot \textbf{v} = 1 (y^{5}) + x (y^{4})+ x^{2}(y^{3})+ x^{3}(y^{2})+ x^{4} (y)+ x^{5 }(1)$

$\textbf{u} \cdot \textbf{v} = y^{5} + x y^{4} + x^{2}y^{3}+ x^{3}y^{2}+ x^{4} y+ x^{5 }$

The degree of a term of a polynomial is the sum of the exponents of its variables. Each term in this polynomial has exponent sum 5, so each term has degree 5. The degree of the polynomial is the greatest of the degrees, so the polynomial has degree 5.

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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 #131 : Matrices

Example Question #132 : Matrices

Example Question #11 : Vector Vector Product

Example Question #12 : Vector Vector Product

Example Question #13 : Vector Vector Product

Example Question #14 : Vector Vector Product

Example Question #831 : Linear Algebra

Example Question #16 : Vector Vector Product

Example Question #17 : Vector Vector Product

Example Question #11 : Vector Vector Product

All Linear Algebra Resources

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