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Calculus 3 : Vectors and Vector Operations

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Example Questions

Example Question #113 : Dot Product

Let a = (1,−2,1), b = (2,−3,−2), and c = (2,0,4). Then $(b\cdot c)a=$

Possible Answers:

(12,0,24)

Cannot be determined

$4\sqrt{6}$

(-4,8,-4)

Correct answer:

(-4,8,-4)

Explanation:

Lets start by $\vec{b}\cdot \vec{c}$ :

$\vec{b}\cdot \vec{c}=4+0-8=-4$

Then we multiply $-4\vec{a}$ :

$-4\vec{a}=(-4,8,-4)$

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Example Question #117 : Dot Product

Find all numbers x for which 2i+5j+2xk ⊥ 6i+4j−xk:

Possible Answers:

$x=\pm1$

x=0,4

x=1

x=4

$x=\pm4$

Correct answer:

$x=\pm4$

Explanation:

if 2i+5j+2xk ⊥ 6i+4j−xk, then the dot product of the two vectors should be 0.

$(2,5,2x)\cdot (6,4,-x)=0$

12+20-2x^2=0

x^2=16

Therefore,

$x=\pm4$

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Example Question #114 : Dot Product

Find the dot product between the two vectors

$u=<5\sqrt3, 5>$

$v=<-2,2\sqrt3>$

Possible Answers:

$50\sqrt3$

225

Correct answer:

Explanation:

The dot product for the vectors

a=<a_1,a_2>

b=<b_1,b_2>

is defined as

$a\bullet b=a_1*b_1+a_2*b_2$

For the vectors in this problem we find that

$u\bullet v=5\sqrt3(-2)+5(2\sqrt3)=-10\sqrt3+10\sqrt3=0$

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Example Question #115 : Dot Product

Find the dot product between $\left \langle 3,-6,1\right \rangle$ and $\left \langle 1,4,5\right \rangle$

Possible Answers:

$\left \langle 3,-24,5\right \rangle$

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 in the problem statement, this becomes (3*1)+(-6*1)+(1*5)=-16 .

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Example Question #120 : Dot Product

Evaluate the dot product $u\bullet v$ .

u=<a,b>

v=<1,0>

Possible Answers:

a+b

Correct answer:

Explanation:

The dot product for the vectors

r=<a_1,a_2>

s=<b_1,b_2>

is defined as

$r\bullet s=a_1*b_1+a_2*b_2$

For the vectors in this problem we find that

$r\bullet s=a(1)+b(0)=a$

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Example Question #121 : Dot Product

Find the dot product between $\left \langle 5,2,7\right \rangle$ and $\left \langle 2,-1,3\right \rangle$

Possible Answers:

Correct answer:

Explanation:

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Example Question #431 : Vectors And Vector Operations

Solve:

$\left \langle x, x^2+y^2\right \rangle \cdot \left \langle x+1, 5\right \rangle$

Possible Answers:

5x^2+x+5y^2

2x+6+x^2+y^2

6x^2+x+5y^2

-4x^2+x-5y^2

Correct answer:

6x^2+x+5y^2

Explanation:

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

Using this, we get

x^2+x+5x^2+5y^2=6x^2+x+5y^2

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Example Question #122 : Dot Product

Find the dot product between the vectors $\left \langle 5,20,8\right \rangle$ and $\left \langle 3,-1,-9\right \rangle$ .

Possible Answers:

$\left \langle 15,-20,-72\right \rangle$

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 $a\cdot b=\left ( x_1*x_2\right )+(y_1*y_2)+(z_1*z_2)$ . We then get $\left ( 5*3\right )+(20*-1)+(8*-9)=(15)+(-20)+(-72)=-77$

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Example Question #432 : Vectors And Vector Operations

Solve:

$\left \langle \sec(z^2), 4, x \right \rangle \cdot \left \langle 5, xy, 3y\right \rangle$

Possible Answers:

$5\sec(z^2)+12xy$

$5\sec(z^2)+4x+3xy$

$\left \langle 5\sec(z^2), 7xy \right \rangle$

$5\sec(z^2)+7xy$

Correct answer:

$5\sec(z^2)+7xy$

Explanation:

The dot product of two vectors is given by the sum of the products of the corresponding components (for example $\left \langle a, b, c\right \rangle \cdot \left \langle e, f, g\right \rangle = ae+bf+cg$ )

Our final answer is

$5\sec(z^2)+7xy$

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Example Question #123 : Dot Product

Find the dot product between the vectors $\left \langle 11,-10,8\right \rangle$ and $\left \langle -1,2,-3\right \rangle$ .

Possible Answers:

Correct answer:

Explanation:

he 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 $a\cdot b=\left ( x_1*x_2\right )+(y_1*y_2)+(z_1*z_2)$ . We then get $\left ( 11*-1\right )+(-10*2)+(8*-3)=(-11)+(-20)+(-24)=-55$

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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 #113 : Dot Product

Example Question #117 : Dot Product

Example Question #114 : Dot Product

Example Question #115 : Dot Product

Example Question #120 : Dot Product

Example Question #121 : Dot Product

Example Question #431 : Vectors And Vector Operations

Example Question #122 : Dot Product

Example Question #432 : Vectors And Vector Operations

Example Question #123 : Dot Product

All Calculus 3 Resources

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