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Calculus 3 : Vectors and Vector Operations

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Example Questions

Example Question #141 : Vectors And Vector Operations

Determine the cross product $\overrightarrow{a}\times\overrightarrow{b}$ , if $\overrightarrow{a}=12\widehat{i}-11\widehat{j}$ and $\overrightarrow{b}=8\widehat{i}+7\widehat{j}$

Possible Answers:

$64\widehat{k}$

$-4\widehat{k}$

$92\widehat{k}$

$172\widehat{k}$

$4\widehat{k}$

Correct answer:

$172\widehat{k}$

Explanation:

The cross product of two vectors is a new vector. In order to find the cross product, it is useful to know the following relationships between the directional vectors:

$\widehat{i}\times\widehat{j}=\widehat{k};\widehat{j}\times\widehat{i}=-\widehat{k}$

$\widehat{j}\times\widehat{k}=\widehat{i};\widehat{k}\times\widehat{j}=-\widehat{i}$

$\widehat{k}\times\widehat{i}=\widehat{j};\widehat{i}\times\widehat{k}=-\widehat{j}$

$\widehat{i}\times\widehat{i}=\widehat{j}\times\widehat{j}=\widehat{k}\times\widehat{k}=0$

Note how the sign changes when terms are reordered; order matters!

Scalar values (the numerical coefficients) multiply through, e.g:

$r\widehat{i}\times s\widehat{j}=(rs)\widehat{k}$

With these principles in mind, we can calculate the cross product of our vectors $\overrightarrow{a}=12\widehat{i}-11\widehat{j}$ and $\overrightarrow{b}=8\widehat{i}+7\widehat{j}$

$\overrightarrow{a}\times \overrightarrow{b}=(0+84-(-88)+0)\widehat{k}=172\widehat{k}$

Report an Error

Example Question #142 : Vectors And Vector Operations

Determine the cross product $\overrightarrow{a}\times\overrightarrow{b}$ , if $\overrightarrow{a}=16\widehat{i}+9\widehat{j}$ and $\overrightarrow{b}=4\widehat{i}+3\widehat{j}$

Possible Answers:

$84\widehat{k}$

$12\widehat{k}$

$37\widehat{k}$

$91\widehat{k}$

Correct answer:

$12\widehat{k}$

Explanation:

The cross product of two vectors is a new vector. In order to find the cross product, it is useful to know the following relationships between the directional vectors:

$\widehat{i}\times\widehat{j}=\widehat{k};\widehat{j}\times\widehat{i}=-\widehat{k}$

$\widehat{j}\times\widehat{k}=\widehat{i};\widehat{k}\times\widehat{j}=-\widehat{i}$

$\widehat{k}\times\widehat{i}=\widehat{j};\widehat{i}\times\widehat{k}=-\widehat{j}$

$\widehat{i}\times\widehat{i}=\widehat{j}\times\widehat{j}=\widehat{k}\times\widehat{k}=0$

Note how the sign changes when terms are reordered; order matters!

Scalar values (the numerical coefficients) multiply through, e.g:

$r\widehat{i}\times s\widehat{j}=(rs)\widehat{k}$

With these principles in mind, we can calculate the cross product of our vectors $\overrightarrow{a}=16\widehat{i}+9\widehat{j}$ and $\overrightarrow{b}=4\widehat{i}+3\widehat{j}$

$\overrightarrow{a}\times \overrightarrow{b}=(0+48-36+0)\widehat{k}=12\widehat{k}$

Report an Error

Example Question #143 : Vectors And Vector Operations

Determine the cross product $\overrightarrow{a}\times\overrightarrow{b}$ , if $\overrightarrow{a}=\widehat{i}+17\widehat{j}$ and $\overrightarrow{b}=\widehat{i}+7\widehat{j}$

Possible Answers:

$14\widehat{k}$

$17\widehat{k}$

$24\widehat{k}$

$-10\widehat{k}$

$7\widehat{k}$

Correct answer:

$-10\widehat{k}$

Explanation:

The cross product of two vectors is a new vector. In order to find the cross product, it is useful to know the following relationships between the directional vectors:

$\widehat{i}\times\widehat{j}=\widehat{k};\widehat{j}\times\widehat{i}=-\widehat{k}$

$\widehat{j}\times\widehat{k}=\widehat{i};\widehat{k}\times\widehat{j}=-\widehat{i}$

$\widehat{k}\times\widehat{i}=\widehat{j};\widehat{i}\times\widehat{k}=-\widehat{j}$

$\widehat{i}\times\widehat{i}=\widehat{j}\times\widehat{j}=\widehat{k}\times\widehat{k}=0$

Note how the sign changes when terms are reordered; order matters!

Scalar values (the numerical coefficients) multiply through, e.g:

$r\widehat{i}\times s\widehat{j}=(rs)\widehat{k}$

With these principles in mind, we can calculate the cross product of our vectors $\overrightarrow{a}=\widehat{i}+17\widehat{j}$ and $\overrightarrow{b}=\widehat{i}+7\widehat{j}$

$\overrightarrow{a}\times \overrightarrow{b}=(0+7-17+0)\widehat{k}=-10\widehat{k}$

Report an Error

Example Question #144 : Vectors And Vector Operations

Determine the cross product $\overrightarrow{a}\times\overrightarrow{b}$ , if $\overrightarrow{a}=21\widehat{i}-9\widehat{j}$ and $\overrightarrow{b}=-3\widehat{i}+\widehat{j}$

Possible Answers:

$-48\widehat{k}$

$-6\widehat{k}$

$48\widehat{k}$

$6\widehat{k}$

Correct answer:

$-6\widehat{k}$

Explanation:

The cross product of two vectors is a new vector. In order to find the cross product, it is useful to know the following relationships between the directional vectors:

$\widehat{i}\times\widehat{j}=\widehat{k};\widehat{j}\times\widehat{i}=-\widehat{k}$

$\widehat{j}\times\widehat{k}=\widehat{i};\widehat{k}\times\widehat{j}=-\widehat{i}$

$\widehat{k}\times\widehat{i}=\widehat{j};\widehat{i}\times\widehat{k}=-\widehat{j}$

$\widehat{i}\times\widehat{i}=\widehat{j}\times\widehat{j}=\widehat{k}\times\widehat{k}=0$

Note how the sign changes when terms are reordered; order matters!

Scalar values (the numerical coefficients) multiply through, e.g:

$r\widehat{i}\times s\widehat{j}=(rs)\widehat{k}$

With these principles in mind, we can calculate the cross product of our vectors $\overrightarrow{a}=21\widehat{i}-9\widehat{j}$ and $\overrightarrow{b}=-3\widehat{i}+\widehat{j}$

$\overrightarrow{a}\times \overrightarrow{b}=(0+21-27+0)\widehat{k}=-6\widehat{k}$

Report an Error

Example Question #145 : Vectors And Vector Operations

Determine the cross product $\overrightarrow{a}\times\overrightarrow{b}$ , if $\overrightarrow{a}=11\widehat{i}+10\widehat{j}$ and $\overrightarrow{b}=-4\widehat{i}-3\widehat{j}$

Possible Answers:

$-73\widehat{k}$

$-7\widehat{k}$

$73\widehat{k}$

$7\widehat{k}$

$-74\widehat{k}$

Correct answer:

$7\widehat{k}$

Explanation:

The cross product of two vectors is a new vector. In order to find the cross product, it is useful to know the following relationships between the directional vectors:

$\widehat{i}\times\widehat{j}=\widehat{k};\widehat{j}\times\widehat{i}=-\widehat{k}$

$\widehat{j}\times\widehat{k}=\widehat{i};\widehat{k}\times\widehat{j}=-\widehat{i}$

$\widehat{k}\times\widehat{i}=\widehat{j};\widehat{i}\times\widehat{k}=-\widehat{j}$

$\widehat{i}\times\widehat{i}=\widehat{j}\times\widehat{j}=\widehat{k}\times\widehat{k}=0$

Note how the sign changes when terms are reordered; order matters!

Scalar values (the numerical coefficients) multiply through, e.g:

$r\widehat{i}\times s\widehat{j}=(rs)\widehat{k}$

With these principles in mind, we can calculate the cross product of our vectors $\overrightarrow{a}=11\widehat{i}+10\widehat{j}$ and $\overrightarrow{b}=-4\widehat{i}-3\widehat{j}$

$\overrightarrow{a}\times \overrightarrow{b}=(0-33-(-40)+0)\widehat{k}=7\widehat{k}$

Report an Error

Example Question #146 : Vectors And Vector Operations

Determine the cross product $\overrightarrow{a}\times\overrightarrow{b}$ , if $\overrightarrow{a}=16\widehat{i}-15\widehat{j}$ and $\overrightarrow{b}=\widehat{j}-\widehat{i}$

Possible Answers:

$\widehat{k}$

$-\widehat{k}$

$-31\widehat{k}$

$31\widehat{k}$

Correct answer:

$\widehat{k}$

Explanation:

The cross product of two vectors is a new vector. In order to find the cross product, it is useful to know the following relationships between the directional vectors:

$\widehat{i}\times\widehat{j}=\widehat{k};\widehat{j}\times\widehat{i}=-\widehat{k}$

$\widehat{j}\times\widehat{k}=\widehat{i};\widehat{k}\times\widehat{j}=-\widehat{i}$

$\widehat{k}\times\widehat{i}=\widehat{j};\widehat{i}\times\widehat{k}=-\widehat{j}$

$\widehat{i}\times\widehat{i}=\widehat{j}\times\widehat{j}=\widehat{k}\times\widehat{k}=0$

Note how the sign changes when terms are reordered; order matters!

Scalar values (the numerical coefficients) multiply through, e.g:

$r\widehat{i}\times s\widehat{j}=(rs)\widehat{k}$

With these principles in mind, we can calculate the cross product of our vectors $\overrightarrow{a}=16\widehat{i}-15\widehat{j}$ and $\overrightarrow{b}=\widehat{j}-\widehat{i}$

$\overrightarrow{a}\times \overrightarrow{b}=(16+0+0-15)\widehat{k}=\widehat{k}$

Report an Error

Example Question #147 : Vectors And Vector Operations

Determine the cross product $\overrightarrow{a}\times\overrightarrow{b}$ , if $\overrightarrow{a}=11\widehat{i}+8\widehat{j}$ and $\overrightarrow{b}=2\widehat{i}+5\widehat{j}$

Possible Answers:

$39\widehat{k}$

$-71\widehat{k}$

$62\widehat{k}$

$71\widehat{k}$

$-18\widehat{k}$

Correct answer:

$39\widehat{k}$

Explanation:

The cross product of two vectors is a new vector. In order to find the cross product, it is useful to know the following relationships between the directional vectors:

$\widehat{i}\times\widehat{j}=\widehat{k};\widehat{j}\times\widehat{i}=-\widehat{k}$

$\widehat{j}\times\widehat{k}=\widehat{i};\widehat{k}\times\widehat{j}=-\widehat{i}$

$\widehat{k}\times\widehat{i}=\widehat{j};\widehat{i}\times\widehat{k}=-\widehat{j}$

$\widehat{i}\times\widehat{i}=\widehat{j}\times\widehat{j}=\widehat{k}\times\widehat{k}=0$

Note how the sign changes when terms are reordered; order matters!

Scalar values (the numerical coefficients) multiply through, e.g:

$r\widehat{i}\times s\widehat{j}=(rs)\widehat{k}$

With these principles in mind, we can calculate the cross product of our vectors $\overrightarrow{a}=11\widehat{i}+8\widehat{j}$ and $\overrightarrow{b}=2\widehat{i}+5\widehat{j}$

$\overrightarrow{a}\times \overrightarrow{b}=(0+55-16+0)\widehat{k}=39\widehat{k}$

Report an Error

Example Question #148 : Vectors And Vector Operations

Determine the cross product $\overrightarrow{a}\times\overrightarrow{b}$ , if $\overrightarrow{a}=117\widehat{i}+3\widehat{j}$ and $\overrightarrow{b}=39\widehat{i}+\widehat{j}$

Possible Answers:

$-39\widehat{k}$

$39\widehat{k}$

$234\widehat{k}$

$-234\widehat{k}$

Correct answer:

Explanation:

The cross product of two vectors is a new vector. In order to find the cross product, it is useful to know the following relationships between the directional vectors:

$\widehat{i}\times\widehat{j}=\widehat{k};\widehat{j}\times\widehat{i}=-\widehat{k}$

$\widehat{j}\times\widehat{k}=\widehat{i};\widehat{k}\times\widehat{j}=-\widehat{i}$

$\widehat{k}\times\widehat{i}=\widehat{j};\widehat{i}\times\widehat{k}=-\widehat{j}$

$\widehat{i}\times\widehat{i}=\widehat{j}\times\widehat{j}=\widehat{k}\times\widehat{k}=0$

Note how the sign changes when terms are reordered; order matters!

Scalar values (the numerical coefficients) multiply through, e.g:

$r\widehat{i}\times s\widehat{j}=(rs)\widehat{k}$

With these principles in mind, we can calculate the cross product of our vectors $\overrightarrow{a}=117\widehat{i}+3\widehat{j}$ and $\overrightarrow{b}=39\widehat{i}+\widehat{j}$

$\overrightarrow{a}\times \overrightarrow{b}=(0+117-117+0)\widehat{k}=0$

The zero result means these two vectors must be parallel.

Report an Error

Example Question #149 : Vectors And Vector Operations

Determine the cross product $\overrightarrow{a}\times\overrightarrow{b}$ , if $\overrightarrow{a}=43\widehat{i}+11\widehat{j}$ and $\overrightarrow{b}=4\widehat{i}+\widehat{j}$

Possible Answers:

$-87\widehat{k}$

$-\widehat{k}$

$87\widehat{k}$

$\widehat{k}$

Correct answer:

$-\widehat{k}$

Explanation:

The cross product of two vectors is a new vector. In order to find the cross product, it is useful to know the following relationships between the directional vectors:

$\widehat{i}\times\widehat{j}=\widehat{k};\widehat{j}\times\widehat{i}=-\widehat{k}$

$\widehat{j}\times\widehat{k}=\widehat{i};\widehat{k}\times\widehat{j}=-\widehat{i}$

$\widehat{k}\times\widehat{i}=\widehat{j};\widehat{i}\times\widehat{k}=-\widehat{j}$

$\widehat{i}\times\widehat{i}=\widehat{j}\times\widehat{j}=\widehat{k}\times\widehat{k}=0$

Note how the sign changes when terms are reordered; order matters!

Scalar values (the numerical coefficients) multiply through, e.g:

$r\widehat{i}\times s\widehat{j}=(rs)\widehat{k}$

With these principles in mind, we can calculate the cross product of our vectors $\overrightarrow{a}=43\widehat{i}+11\widehat{j}$ and $\overrightarrow{b}=4\widehat{i}+\widehat{j}$

$\overrightarrow{a}\times \overrightarrow{b}=(0+43-44+0)\widehat{k}=-\widehat{k}$

Report an Error

Example Question #150 : Vectors And Vector Operations

Determine the cross product $\overrightarrow{a}\times\overrightarrow{b}$ , if $\overrightarrow{a}=44\widehat{i}-2\widehat{j}$ and $\overrightarrow{b}=-23\widehat{i}+\widehat{j}$

Possible Answers:

$90\widehat{k}$

$-8\widehat{k}$

$-2\widehat{k}$

$-90\widehat{k}$

Correct answer:

$-2\widehat{k}$

Explanation:

The cross product of two vectors is a new vector. In order to find the cross product, it is useful to know the following relationships between the directional vectors:

$\widehat{i}\times\widehat{j}=\widehat{k};\widehat{j}\times\widehat{i}=-\widehat{k}$

$\widehat{j}\times\widehat{k}=\widehat{i};\widehat{k}\times\widehat{j}=-\widehat{i}$

$\widehat{k}\times\widehat{i}=\widehat{j};\widehat{i}\times\widehat{k}=-\widehat{j}$

$\widehat{i}\times\widehat{i}=\widehat{j}\times\widehat{j}=\widehat{k}\times\widehat{k}=0$

Note how the sign changes when terms are reordered; order matters!

Scalar values (the numerical coefficients) multiply through, e.g:

$r\widehat{i}\times s\widehat{j}=(rs)\widehat{k}$

With these principles in mind, we can calculate the cross product of our vectors $\overrightarrow{a}=44\widehat{i}-2\widehat{j}$ and $\overrightarrow{b}=-23\widehat{i}+\widehat{j}$

$\overrightarrow{a}\times \overrightarrow{b}=(0+44-46+0)\widehat{k}=-2\widehat{k}$

Report an Error

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Calculus 3 : Vectors and Vector Operations

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #141 : Vectors And Vector Operations

Example Question #142 : Vectors And Vector Operations

Example Question #143 : Vectors And Vector Operations

Example Question #144 : Vectors And Vector Operations

Example Question #145 : Vectors And Vector Operations

Example Question #146 : Vectors And Vector Operations

Example Question #147 : Vectors And Vector Operations

Example Question #148 : Vectors And Vector Operations

Example Question #149 : Vectors And Vector Operations

Example Question #150 : Vectors And Vector Operations

All Calculus 3 Resources

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