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Calculus 3 : Calculus 3

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #421 : Vectors And Vector Operations

Compute $a\cdot b$ , where $a=\left \langle 3,0,4\right \rangle$ and $b=\left \langle 9,11,2\right \rangle$ .

Possible Answers:

$\left \langle 2,11,2\right \rangle$

$\left \langle -6,-11,4\right \rangle$

Correct answer:

Explanation:

The formula for the dot product is

$a\cdot b=(i_1*i_2)+(j_1*j_2)+(k_1*k_2)$ .

Using the given vectors, we get

(3*9)+(0*11)+(4*2)=35 .

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Example Question #2432 : Calculus 3

Compute the dot product of the vectors $\left \langle 2,9,21\right \rangle$ and $\left \langle 2,5,0\right \rangle$ .

Possible Answers:

$\left \langle 4,45,0\right \rangle$

Correct answer:

Explanation:

The formula for the dot product of vectors

$a=\left \langle x_1,y_1,z_1\right \rangle$

and

$b=\left \langle x_2,y_2,z_2\right \rangle$

$a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$ .

Using the vectors we were given, this becomes

(2*2)+(9*5)+(21*0) which equals

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Example Question #2433 : Calculus 3

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

Possible Answers:

(-4,8,-4)

$4\sqrt{6}$

Cannot be determined

(12,0,24)

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 #112 : Dot Product

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

Possible Answers:

$x=\pm1$

x=4

x=0,4

$x=\pm4$

x=1

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 #2434 : Calculus 3

Find the dot product between the two vectors

$u=<5\sqrt3, 5>$

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

Possible Answers:

225

$50\sqrt3$

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 #431 : Vectors And Vector Operations

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 #432 : Vectors And Vector Operations

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 #2437 : Calculus 3

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 #2431 : Calculus 3

Solve:

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

Possible Answers:

6x^2+x+5y^2

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

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

5x^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 #2439 : Calculus 3

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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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #421 : Vectors And Vector Operations

Example Question #2432 : Calculus 3

Example Question #2433 : Calculus 3

Example Question #112 : Dot Product

Example Question #2434 : Calculus 3

Example Question #431 : Vectors And Vector Operations

Example Question #432 : Vectors And Vector Operations

Example Question #2437 : Calculus 3

Example Question #2431 : Calculus 3

Example Question #2439 : Calculus 3

All Calculus 3 Resources

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