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Calculus 3 : Dot Product

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Example Questions

Example Question #111 : Dot Product

Solve:

$\left \langle 3\cos^2(x), 5e^y\right \rangle \cdot \left \langle 4\sin^2(x), 1\right \rangle$ $\displaystyle \left \langle 3\cos^2(x), 5e^y\right \rangle \cdot \left \langle 4\sin^2(x), 1\right \rangle$

Possible Answers:

$\left \langle12\cos^2(x)\sin^2(x), 5e^y \right \rangle$ $\displaystyle \left \langle12\cos^2(x)\sin^2(x), 5e^y \right \rangle$

12+5e^y $\displaystyle 12+5e^y$

$12\cos^2(x)\sin^2(x)+e^y$ $\displaystyle 12\cos^2(x)\sin^2(x)+e^y$

$12\cos^2(x)\sin^2(x)+5e^y$ $\displaystyle 12\cos^2(x)\sin^2(x)+5e^y$

Correct answer:

$12\cos^2(x)\sin^2(x)+5e^y$ $\displaystyle 12\cos^2(x)\sin^2(x)+5e^y$

Explanation:

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

Our answer is

$12\cos^2(x)\sin^2(x)+5e^y$ $\displaystyle 12\cos^2(x)\sin^2(x)+5e^y$

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Example Question #427 : Vectors And Vector Operations

Find the dot product.

u=<2,7> $\displaystyle u=< 2,7>$

v=<5,10> $\displaystyle v=< 5,10>$

Possible Answers:

$\displaystyle 80$

240 $\displaystyle 240$

$\displaystyle 36$

700 $\displaystyle 700$

Correct answer:

$\displaystyle 80$

Explanation:

The dot product for the vectors

a=<a_1,a_2> $\displaystyle a=< a_1,a_2>$

b=<b_1,b_2> $\displaystyle b=< b_1,b_2>$

is defined as

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

For the vectors in this problem we find that

$u\bullet v=2(5)+7(10)=10+70=80$ $\displaystyle u\bullet v=2(5)+7(10)=10+70=80$

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Example Question #428 : Vectors And Vector Operations

Find the dot product.

u=<1,0> $\displaystyle u=< 1,0>$

v=<3,2> $\displaystyle v=< 3,2>$

Possible Answers:

$\displaystyle 10$

$\displaystyle 3$

$\displaystyle 0$

$\displaystyle 6$

Correct answer:

$\displaystyle 3$

Explanation:

The dot product for the vectors

a=<a_1,a_2> $\displaystyle a=< a_1,a_2>$

b=<b_1,b_2> $\displaystyle b=< b_1,b_2>$

is defined as

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

For the vectors in this problem we find that

$u\bullet v=1(3)+0(2)=3+0=3$ $\displaystyle u\bullet v=1(3)+0(2)=3+0=3$

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Example Question #112 : Dot Product

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

Possible Answers:

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

$\displaystyle 35$

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

$\displaystyle 27$

Correct answer:

$\displaystyle 35$

Explanation:

The formula for the dot product is

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

Using the given vectors, we get

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

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Example Question #113 : Dot Product

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

Possible Answers:

$\displaystyle 3$

$\displaystyle 7$

$\displaystyle 49$

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

Correct answer:

$\displaystyle 49$

Explanation:

The formula for the dot product of vectors

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

and

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

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

Using the vectors we were given, this becomes

$\displaystyle (2*2)+(9*5)+(21*0)$ which equals $\displaystyle 49$

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Example Question #116 : Dot Product

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

Possible Answers:

Cannot be determined

$\displaystyle (12,0,24)$

$4\sqrt{6}$ $\displaystyle 4\sqrt{6}$

$\displaystyle (-4,8,-4)$

$\displaystyle 4$

Correct answer:

$\displaystyle (-4,8,-4)$

Explanation:

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

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

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

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

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Example Question #117 : Dot Product

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

Possible Answers:

x=4 $\displaystyle x=4$

$x=\pm4$ $\displaystyle x=\pm4$

x=0,4 $\displaystyle x=0,4$

$x=\pm1$ $\displaystyle x=\pm1$

x=1 $\displaystyle x=1$

Correct answer:

$x=\pm4$ $\displaystyle x=\pm4$

Explanation:

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

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

$\displaystyle 12+20-2x^2=0$

x^2=16 $\displaystyle x^2=16$

Therefore,

$x=\pm4$ $\displaystyle x=\pm4$

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Example Question #118 : Dot Product

Find the dot product between the two vectors

$u=<5\sqrt3, 5>$ $\displaystyle u=< 5\sqrt3, 5>$

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

Possible Answers:

225 $\displaystyle 225$

$\displaystyle 0$

$\displaystyle -300$

$50\sqrt3$ $\displaystyle 50\sqrt3$

Correct answer:

$\displaystyle 0$

Explanation:

The dot product for the vectors

a=<a_1,a_2> $\displaystyle a=< a_1,a_2>$

b=<b_1,b_2> $\displaystyle b=< b_1,b_2>$

is defined as

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

For the vectors in this problem we find that

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

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Example Question #119 : Dot Product

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

Possible Answers:

$\displaystyle 8$

$\displaystyle 16$

$\displaystyle -16$

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

Correct answer:

$\displaystyle -16$

Explanation:

The formula for the dot product between two vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ $\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ $\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle$ is $a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$ $\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$. Using the vectors in the problem statement, this becomes $\displaystyle (3*1)+(-6*1)+(1*5)=-16$.

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Example Question #112 : Dot Product

Evaluate the dot product $u\bullet v$ $\displaystyle u\bullet v$.

u=<a,b> $\displaystyle u=< a,b>$

v=<1,0> $\displaystyle v=< 1,0>$

Possible Answers:

a+b $\displaystyle a+b$

$\displaystyle b$

$\displaystyle 0$

$\displaystyle a$

Correct answer:

$\displaystyle a$

Explanation:

The dot product for the vectors

r=<a_1,a_2> $\displaystyle r=< a_1,a_2>$

s=<b_1,b_2> $\displaystyle s=< b_1,b_2>$

is defined as

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

For the vectors in this problem we find that

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

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Calculus 3 : Dot Product

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #111 : Dot Product

Example Question #427 : Vectors And Vector Operations

Example Question #428 : Vectors And Vector Operations

Example Question #112 : Dot Product

Example Question #113 : Dot Product

Example Question #116 : Dot Product

Example Question #117 : Dot Product

Example Question #118 : Dot Product

Example Question #119 : Dot Product

Example Question #112 : Dot Product

All Calculus 3 Resources

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