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Calculus 3 : Vectors and Vector Operations

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Example Questions

Example Question #481 : Vectors And Vector Operations

Find the dot product between the vectors $\left \langle -1,7,-2\right \rangle$ and $\left \langle 6,1,5\right \rangle$

Possible Answers:

Correct answer:

Explanation:

To find the dot product between the vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , we use the formula:

$a\cdot b=(a_1b_1)+(a_2b_2)+(a_3b_3)$

Using the vectors from the problem statement and applying, we get

(-1*6)+(7*1)+(-2*5)=-9

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Example Question #482 : Vectors And Vector Operations

Find the dot product between the vectors $\left \langle -5,0,2\right \rangle$ and $\left \langle 10,2,7\right \rangle$

Possible Answers:

Correct answer:

Explanation:

To find the dot product between two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , you apply the formula:

$a\cdot b=a_1b_1+a_2b_2+a_3b_3$

Using the vectors from the problem statement, we get

(-5*10)+(0*2)+(2*7)=-36

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Example Question #483 : Vectors And Vector Operations

Find the dot product between the vectors $\left \langle -4,-3,6\right \rangle$ and $\left \langle 2,7,-1\right \rangle$

Possible Answers:

Correct answer:

Explanation:

To find the dot product between two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , you apply the formula:

$a\cdot b=a_1b_1+a_2b_2+a_3b_3$

Using the vectors from the problem statement, we get

(-4*2)+(-3*7)+(6*-1)=-35

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Example Question #481 : Vectors And Vector Operations

Solve:

$\left \langle -5, 4, 2\right \rangle \cdot \left \langle 2x, xy, z^3\right \rangle$

Possible Answers:

-10x+4xy+2z^3

10x+4xy+2z^3

$\left \langle -10x,4xy,2z^3 \right \rangle$

-80x^2yz^3

Correct answer:

-10x+4xy+2z^3

Explanation:

The dot product is given by the sum of the products of the corresponding components of each vector (for example, $\left \langle a,b\right \rangle\cdot \left \langle x,y\right \rangle= ax+by$ )

Our final answer is

-10x+4xy+2z^3

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Example Question #482 : Vectors And Vector Operations

Solve for x:

$\left \langle x, 2x, 5\right \rangle \cdot \left \langle 3, 1, 0\right \rangle =4$

Possible Answers:

$x=\frac{1}{5}$

x=0

$x=\frac{4}{5}$

$x=\frac{5}{4}$

Correct answer:

$x=\frac{4}{5}$

Explanation:

To solve for x, we must take the dot product, which is performed by multiplying the corresponding components and adding the products together.

We get

3x+2x+0=4

Solving for x, we get

5x=4

$x=\frac{4}{5}$

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Example Question #486 : Vectors And Vector Operations

Find the dot product of the vectors $\left \langle 2,-4,1\right \rangle$ and $\left \langle -1,3,6\right \rangle$

Possible Answers:

Correct answer:

Explanation:

To find the dot product of two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , we apply the formula $a\cdot b=a_1b_1+a_2b_2+a_3b_3$

Using the vectors from the problem statement, we get

(2*-1)+(-4*3)+(6*1)=-8

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Example Question #487 : Vectors And Vector Operations

Find the dot product of the vectors $\left \langle -2,-5,-8\right \rangle$ and $\left \langle 2,-9,1\right \rangle$

Possible Answers:

Correct answer:

Explanation:

To find the dot product of two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , we apply the formula $a\cdot b=a_1b_1+a_2b_2+a_3b_3$

Using the vectors from the problem statement, we get

(2*-2)+(-5*-9)+(-8*1)=33

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Example Question #488 : Vectors And Vector Operations

Find the dot product of the vectors $\left \langle 2,0,-4\right \rangle$ and $\left \langle 9,4,1\right \rangle$

Possible Answers:

Correct answer:

Explanation:

To find the dot product of two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , we use the formula $a\cdot b=a_1b_1+a_2b_2+a_3b_3$

Using the vectors from the problem statement, we get

(2*9)+(0*4)+(-4*1)=14

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Example Question #489 : Vectors And Vector Operations

Find the dot product of the vectors $\left \langle 8,9,5\right \rangle$ and $\left \langle 0,2,-7\right \rangle$

Possible Answers:

Correct answer:

Explanation:

To find the dot product of two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , we use the formula $a\cdot b=a_1b_1+a_2b_2+a_3b_3$

Using the vectors from the problem statement, we get

(8*0)+(9*2)+(5*-7)=-17

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Example Question #483 : Vectors And Vector Operations

Solve:

$\left \langle x, y, z^2\right \rangle\cdot \left \langle 4, xy, 3y\right \rangle$

Possible Answers:

$\left \langle 4x, xy^2,3yz^2 \right \rangle$

4x+x^2y^2+3yz^2

12x^2y^3z^2

4x+xy^2+3yz^2

Correct answer:

4x+xy^2+3yz^2

Explanation:

The dot product of two vectors is equal to the sum of the product of the corresponding components (for example, $\left \langle a,b\right \rangle \cdot \left \langle c,d\right \rangle = ac+bd$ )

The final answer is

4x+xy^2+3yz^2

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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 #481 : Vectors And Vector Operations

Example Question #482 : Vectors And Vector Operations

Example Question #483 : Vectors And Vector Operations

Example Question #481 : Vectors And Vector Operations

Example Question #482 : Vectors And Vector Operations

Example Question #486 : Vectors And Vector Operations

Example Question #487 : Vectors And Vector Operations

Example Question #488 : Vectors And Vector Operations

Example Question #489 : Vectors And Vector Operations

Example Question #483 : Vectors And Vector Operations

All Calculus 3 Resources

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