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Calculus 3 : Vectors and Vector Operations

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Example Questions

Example Question #441 : Vectors And Vector Operations

Find the dot product between the vectors $\left \langle 3,10,5\right \rangle$ and $\left \langle 1,2,6\right \rangle$

Possible Answers:

$\left \langle 3,20,30\right \rangle$

Correct answer:

Explanation:

The formula for the dot product between two vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ is $a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$ . Using the vectors from the problem statement, we get (3*1)+(10*2)+(6*5)=53

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Example Question #441 : Vectors And Vector Operations

Solve:

$\left \langle \sec(z), 14\right \rangle \cdot \left \langle 3, \tan(z)\right \rangle$

Possible Answers:

$\sec(z)+\tan(z)$

$42\sec(z)\tan(z)$

$3\sec(z)+14\tan(z)$

$\left \langle 3\sec(z), 14\tan(z) \right \rangle$

Correct answer:

$3\sec(z)+14\tan(z)$

Explanation:

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

Our final answer is

$3\sec(z)+14\tan(z)$

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Example Question #443 : Vectors And Vector Operations

Find the dot product between $\left \langle 1,-4,5\right \rangle$ and $\left \langle 2,2,1\right \rangle$

Possible Answers:

$\left \langle 2,-8,5\right \rangle$

Correct answer:

Explanation:

To find the dot product between two vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ we use the formula $a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$ . Using the vectors in the problem statement, we get (1*2)+(-4*2)+(5*1)=-1

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Example Question #444 : Vectors And Vector Operations

Find the dot product between $\left \langle \sin(x),y,5\right \rangle$ and $\left \langle \csc(x),2,4\right \rangle$

Possible Answers:

3y+22

4y+20

2y+21

y+5

Correct answer:

2y+21

Explanation:

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Example Question #445 : Vectors And Vector Operations

Find the dot product between $\left \langle x,2y,z\right \rangle$ and $\left \langle 2,2,1\right \rangle$

Possible Answers:

2x+4y+z

x+3y+2z

x+3y-z

2x-3y-z

Correct answer:

2x+4y+z

Explanation:

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Example Question #446 : Vectors And Vector Operations

Find the dot product between $\left \langle 3,2y,z^5\right \rangle$ and $\left \langle 3x,yz,z^2\right \rangle$

Possible Answers:

9x+2x^2y+z^7

9x+2y^2z+z^7

2x^2+2y^2x-z

9x+2yz+z^7

Correct answer:

9x+2y^2z+z^7

Explanation:

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Example Question #447 : Vectors And Vector Operations

Find the dot product between $\left \langle 3x,4y,z\right \rangle$ and $\left \langle 4,2,7\right \rangle$

Possible Answers:

2x+y-9z

4x-3y+5z

12x+8y+7z

x+4y-3z

Correct answer:

12x+8y+7z

Explanation:

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Example Question #448 : Vectors And Vector Operations

Find the dot product between $\left \langle 2,-3,4\right \rangle$ and $\left \langle 5,-4,1\right \rangle$

Possible Answers:

Correct answer:

Explanation:

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Example Question #449 : Vectors And Vector Operations

Find the dot product between $\left \langle x,-3,4z\right \rangle$ and $\left \langle x,y,z\right \rangle$

Possible Answers:

x^2-5y+5z^2

x^2+6y+4z^2

x^2-3y+4z^2

x^3+y+7z^2

Correct answer:

x^2-3y+4z^2

Explanation:

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Example Question #450 : Vectors And Vector Operations

Consider the vector $\mathbf{v}=\mathbf{i}+7\mathbf{j}+2\mathbf{k}$ .

Which of the following vectors are orthogonal to v?

Possible Answers:

$6\mathbf{i}+-3\mathbf{j}+5\mathbf{k}$

$-\mathbf{i}+3\mathbf{k}$

$4\mathbf{i}-2\mathbf{j}+\mathbf{k}$

$-5\mathbf{i}+2\mathbf{j}+3\mathbf{k}$

$-4\mathbf{i}+3\mathbf{j}-2\mathbf{k}$

Correct answer:

$4\mathbf{i}-2\mathbf{j}+\mathbf{k}$

Explanation:

Two vectors are defined as orthogonal when their dot product is zero.

The dot product of two vectors

$\mathbf{u}=u_x\mathbf{i}+u_y\mathbf{j}+u_z\mathbf{k}$ and $\mathbf{v}=v_x\mathbf{i}+v_y\mathbf{j}+v_z\mathbf{k}$

Is given by the expression:

$\mathbf{u}\cdot\mathbf{v}=u_xv_x+u_yv_y+u_zv_z$

The only vector that satisfies the requirement that the dot product of it and v is zero is $4\mathbf{i}-2\mathbf{j}+\mathbf{k}$ :

$(3\mathbf{i}+7\mathbf{j}+2\mathbf{k})\cdot (4\mathbf{i}-2\mathbf{j}+\mathbf{k}) =(3)(4)+(7)(-2)+(2)(2)=12-14+2=0$

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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 #441 : Vectors And Vector Operations

Example Question #441 : Vectors And Vector Operations

Example Question #443 : Vectors And Vector Operations

Example Question #444 : Vectors And Vector Operations

Example Question #445 : Vectors And Vector Operations

Example Question #446 : Vectors And Vector Operations

Example Question #447 : Vectors And Vector Operations

Example Question #448 : Vectors And Vector Operations

Example Question #449 : Vectors And Vector Operations

Example Question #450 : Vectors And Vector Operations

All Calculus 3 Resources

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