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Calculus 2 : Parametric, Polar, and Vector

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Example Questions

Example Question #254 : Vector

Vector $\vec{v} = (-6,6,9)$ .

Calculate the magnitude, or $\left \| \vec{v} \right \|$ , of $\vec{v}$

Possible Answers:

$\left \| \vec{v} \right \|=\sqrt{153}$

$\left \| \vec{v} \right \|=\sqrt{150}$

$\left \| \vec{v} \right \|=9$

$\left \| \vec{v} \right \|=\sqrt{151}$

Correct answer:

$\left \| \vec{v} \right \|=\sqrt{153}$

Explanation:

Calculating magnitude:

If $\vec{v} = (v_1,v_2,v_3)$

Then the magnitude of $\vec{v}$ is $\sqrt{v_1^2 + v_2^2 + v_3^2}$

Note: Magnitude, length, and norm are synonymous.

$\vec{v} = (-6,6,9)$

$\left \| \vec{v} \right \|=\sqrt{(-6)^2+6^2+9^2}$

$\left \| \vec{v} \right \|=\sqrt{36+36+81}$

$\left \| \vec{v} \right \|=\sqrt{153}$

$\left \| \vec{v} \right \|$ cannot be further reduced, and so $\left \| \vec{v} \right \|=\sqrt{153}$

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Example Question #255 : Vector

Vector $\vec{v} = (sin(x),cos(x),-1)$ .

Calculate the magnitude of $\vec{v}$ , as in $\left \| \vec{v} \right \|$

Possible Answers:

$\left \| \vec{v} \right \|=0$

$\left \| \vec{v} \right \|=2$

$\left \| \vec{v} \right \|=1$

$\left \| \vec{v} \right \|=\sqrt{2}$

Correct answer:

$\left \| \vec{v} \right \|=\sqrt{2}$

Explanation:

Calculating magnitude:

If $\vec{v} = (v_1,v_2,v_3)$

Then the magnitude of $\vec{v}$ is $\sqrt{v_1^2 + v_2^2 + v_3^2}$

Note: Magnitude, length, and norm are synonymous.

Note: sin(x)^2 + cos(x)^2 = 1 , and this is a basic trig identity that you should know.

$\vec{v} = (sin(x),cos(x),-1)$

$\left \| \vec{v} \right \|=\sqrt{sin(x)^2+cos(x)^2+(-1)^2}$

$\left \| \vec{v} \right \|=\sqrt{sin(x)^2+cos(x)^2+1}$

$\left \| \vec{v} \right \|=\sqrt{1 + 1}$

$\left \| \vec{v} \right \|=\sqrt{2}$

$\left \| \vec{v} \right \|$ cannot be further reduced, and so $\left \| \vec{v} \right \|=\sqrt{2}$

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Example Question #256 : Vector

Vector $\vec{v} = (2x^3,3x^{3},\sqrt{12}x^3)$ .

Calculate the magnitude, or $\left \| \vec{v} \right \|$ , of $\vec{v}$

Possible Answers:

$\left \| \vec{v} \right \|=4x^3$

$\left \| \vec{v} \right \|=7x^3$

$\left \| \vec{v} \right \|=6x^3$

$\left \| \vec{v} \right \|=5x^3$

Correct answer:

$\left \| \vec{v} \right \|=5x^3$

Explanation:

Calculating magnitude:

If $\vec{v} = (v_1,v_2,v_3)$

Then the magnitude of $\vec{v}$ is $\sqrt{v_1^2 + v_2^2 + v_3^2}$

Note: Magnitude, length, and norm are synonymous.

$\vec{v} = (2x^3,3x^{3},\sqrt{12}x^3)$

$\left \| \vec{v} \right \|=\sqrt{(2x^3)^2+(3x^3)^2+(\sqrt{12}x^3)^2}$

$\left \| \vec{v} \right \|=\sqrt{4x^6+9x^6+12x^6}$

$\left \| \vec{v} \right \|=\sqrt{25x^6}$

$\left \| \vec{v} \right \|=5x^3$

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Example Question #4 : Vector Form

$\newline\vec{v} =(x^2,-x,5)\newline\vec{u} =(2,3x,11)\newline$

Calculate $\vec{v} + \vec{u}$

Possible Answers:

$\vec{v}+\vec{u}=(x^2,2x,16)$

$\vec{v}+\vec{u}=(x^2 + 2,2x,16)$

$\vec{v}+\vec{u}=(x^2 + 2,0,16)$

$\vec{v}+\vec{u}=(x^2,2x,3x)$

Correct answer:

$\vec{v}+\vec{u}=(x^2 + 2,2x,16)$

Explanation:

Calculate the sum of vectors.

In general,

$\vec{v} = (v_1,v_2,v_3)$

$\vec{u} = (u_1,u_2,u_3)$

$\vec{v} + \vec{u} = (v_1 + u_1, v_2 + u_2,v_3 + u_3)$

Solution:

$\newline\vec{v} =(x^2,-x,5)\newline\vec{u} =(2,3x,11)\newline$

$\vec{v}+\vec{u}=(x^2,-x,5)+(2,3x,11)$

$\vec{v}+\vec{u}=(x^2+2,-x+3x,5+11)$

$\vec{v}+\vec{u}=(x^2 + 2,2x,16)$

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Example Question #257 : Vector

$\newline\vec{v} =(2x+3,3x^2+1,e^x+2)\newline\vec{u} =(-2x+4,-11x^2+2,7e^x+x^2+3)\newline$

Calculate $\vec{v} + \vec{u}$

Possible Answers:

$\vec{v}+\vec{u}=(7,-8x^2+3,e^{8x}+x^2+5)$

$\vec{v}+\vec{u}=(7,-8x^2,8e^x+x^2+5)$

$\vec{v}+\vec{u}=(7,-8x+3,8e^x+x^2+4)$

$\vec{v}+\vec{u}=(7,-8x^2+3,8e^x+x^2+5)$

Correct answer:

$\vec{v}+\vec{u}=(7,-8x^2+3,8e^x+x^2+5)$

Explanation:

Calculate the sum of vectors.

In general,

$\vec{v} = (v_1,v_2,v_3)$

$\vec{u} = (u_1,u_2,u_3)$

$\vec{v} + \vec{u} = (v_1 + u_1, v_2 + u_2,v_3 + u_3)$

Solution:

$\vec{v} =(2x+3,3x^2+1,e^x+2)$

$\vec{u} =(-2x+4,-11x^2+2,7e^x+x^2+3)$

$\vec{v}+\vec{u}=(2x+3,3x^2+1,e^x+2)+(-2x+4,-11x^2+2,7e^x+x^2+3)$

$\vec{v}+\vec{u}=(2x+3+-2x+4,3x^2+1+-11x^2+2,e^x+2+7e^x+x^2+3)$

$\vec{v}+\vec{u}=(7,-8x^2+3,8e^x+x^2+5)$

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Example Question #101 : Vector Calculations

$\newline\vec{v} =(sin(x)+2x,2x^2+3x-3,e^{2x}+3)\newline\vec{u} =(-2x+3,2x^4+4x,e^{3x}+2)\newline$

Calculate $\vec{v} + \vec{u}$

Possible Answers:

$\vec{v}+\vec{u}=(sin(x)+3,2x^4+2x^2+7x-2,e^{2x}+e^{3x}+4)$

$\vec{v}+\vec{u}=(sin(x)+3,2x^4+2x^2+7x-3,5e^{x}+5)$

$\vec{v}+\vec{u}=(sin(x)+3,2x^4+2x^2+7x-3,e^{2x}+e^{3x})$

$\vec{v}+\vec{u}=(sin(x)+3,2x^4+2x^2+7x-3,e^{2x}+e^{3x}+5)$

Correct answer:

$\vec{v}+\vec{u}=(sin(x)+3,2x^4+2x^2+7x-3,e^{2x}+e^{3x}+5)$

Explanation:

Calculate the sum of vectors.

In general,

$\vec{v} = (v_1,v_2,v_3)$

$\vec{u} = (u_1,u_2,u_3)$

$\vec{v} + \vec{u} = (v_1 + u_1, v_2 + u_2,v_3 + u_3)$

Solution:

$\vec{v} =(sin(x)+2x,2x^2+3x-3,e^{2x}+3)$

$\vec{u} =(-2x+3,2x^4+4x,e^{3x}+2)$

$\vec{v}+\vec{u}=(sin(x)+2x,2x^2+3x-3,e^{2x}+3)+(-2x+3,2x^4+4x,e^{3x}+2)$

$\vec{v}+\vec{u}=(sin(x)+2x+-2x+3,2x^2+3x-3+2x^4+4x,e^{2x}+3+e^{3x}+2)$

$\vec{v}+\vec{u}=(sin(x)+3,2x^4+2x^2+7x-3,e^{2x}+e^{3x}+5)$

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Example Question #102 : Vector Calculations

$\newline\vec{v} =(5ln(x),2+2x-3x^2,ln(x)+1)\newline\vec{u} =(4+2x,3,\frac{1}{ln(x)+1})\newline$

Calculate $\vec{v} \cdot \vec{u}$

Possible Answers:

$\vec{v}\cdot\vec{u}=20ln(x)+10xln(x)-9x^2+6x+7$

$\vec{v}\cdot\vec{u}=20ln(x)+10xln(x)-9x^2+6x+6$

$\vec{v}\cdot\vec{u}=20ln(x)+ln(10x)-9x^2+6x+6$

$\vec{v}\cdot\vec{u}=10ln(x)+10xln(x)-9x^2+7$

Correct answer:

$\vec{v}\cdot\vec{u}=20ln(x)+10xln(x)-9x^2+6x+7$

Explanation:

Calculate the dot product of the 2 vectors.

In general,

$\vec{v} = (v_1,v_2,v_3)$

$\vec{u} = (u_1,u_2,u_3)$

$\vec{v} \cdot \vec{u} = v_1 * u_1 + v_2 * u_2 + v_3 * u_3$

Solution:

$\vec{v} =(5ln(x),2+2x-3x^2,ln(x)+1)$

$\vec{u} =(4+2x,3,\frac{1}{ln(x)+1})$

$\vec{v}\cdot\vec{u}=(5ln(x),2+2x-3x^2,ln(x)+1)\cdot(4+2x,3,\frac{1}{ln(x)+1})$

$\vec{v}\cdot\vec{u}=5ln(x)*4+2x+2+2x-3x^2*3+(ln(x)+1)*\frac{1}{ln(x)+1}$

$\vec{v}\cdot\vec{u}=20ln(x)+10xln(x)+6+6x-9x^2+1$

$\vec{v}\cdot\vec{u}=20ln(x)+10xln(x)-9x^2+6x+7$

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Example Question #103 : Vector Calculations

$\newline\vec{v} =((3x^2+3),e^{2x},(sin(x)+2x))\newline\vec{u} =(2x,(4x+2),-1)\newline$

Calculate $\vec{v} \cdot \vec{u}$

Possible Answers:

$\vec{v}\cdot\vec{u}=3x^3+4x+4xe^{2x}+2e^{x}-sin(x)$

$\vec{v}\cdot\vec{u}=6x^2+4x+4xe^{2x}+2e^{2x}+sin(x)$

$\vec{v}\cdot\vec{u}=6x^3+4x+4xe^{2x}+2e^{2x}-sin(x)$

$\vec{v}\cdot\vec{u}=6x^3+4x+4xe^{2x}+2e^{2x}$

Correct answer:

$\vec{v}\cdot\vec{u}=6x^3+4x+4xe^{2x}+2e^{2x}-sin(x)$

Explanation:

Calculate the dot product of the 2 vectors.

In general,

$\vec{v} = (v_1,v_2,v_3)$

$\vec{u} = (u_1,u_2,u_3)$

$\vec{v} \cdot \vec{u} = v_1 * u_1 + v_2 * u_2 + v_3 * u_3$

Solution:

$\vec{v} =((3x^2+3),e^{2x},(sin(x)+2x))$

$\vec{u} =(2x,(4x+2),-1)$

$\vec{v}\cdot\vec{u}=((3x^2+3),e^{2x},(sin(x)+2x)\cdot(2x,(4x+2),-1)$

$\vec{v}\cdot\vec{u}=(3x^2+3)*2x+e^{2x}*(4x+2)+(sin(x)+2x)*-1$

$\vec{v}\cdot\vec{u}=6x^3+6x+4xe^{2x}+2e^{2x}+(-sin(x)-2x)$

$\vec{v}\cdot\vec{u}=6x^3+4x+4xe^{2x}+2e^{2x}-sin(x)$

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Example Question #104 : Vector Calculations

$\newline\vec{v} =(x^2+2x,5x^3-2x+1,2x+6)\newline\vec{u} =(2x^2+2x+4,x^2+x,x^2+x)\newline$

Calculate $\vec{v} + \vec{u}$

Possible Answers:

$\vec{v}+\vec{u}=(3x^2+x,5x^3+x^2+5,x^2+2x+6)$

$\vec{v}+\vec{u}=(4x^2+x,5x^3+2x^2+5,x^2+2x)$

$\vec{v}+\vec{u}=(4x^2,5x^3+2x^2,x^2+3x+6)$

$\vec{v}+\vec{u}=(3x^2+4x+4,5x^3+x^2-x+1,x^2+3x+6)$

Correct answer:

$\vec{v}+\vec{u}=(3x^2+4x+4,5x^3+x^2-x+1,x^2+3x+6)$

Explanation:

Calculate the sum of vectors.

In general,

$\vec{v} = (v_1,v_2,v_3)$

$\vec{u} = (u_1,u_2,u_3)$

$\vec{v} + \vec{u} = (v_1 + u_1, v_2 + u_2,v_3 + u_3)$

Solution:

$\vec{v} =(x^2+2x,5x^3-2x+1,2x+6)$

$\vec{u} =(2x^2+2x+4,x^2+x,x^2+x)$

$\vec{v}+\vec{u}=(x^2+2x,5x^3-2x+1,2x+6)+(2x^2+2x+4,x^2+x,x^2+x)$

$\vec{v}+\vec{u}=(x^2+2x+2x^2+2x+4,5x^3-2x+1+x^2+x,2x+6+x^2+x)$

$\vec{v}+\vec{u}=(3x^2+4x+4,5x^3+x^2-x+1,x^2+3x+6)$

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Example Question #105 : Vector Calculations

$\newline\vec{v} =(5x,16+x,2x^2)\newline\vec{u} =(e^x,2,3x)\newline$

Calculate $\vec{v} \cdot \vec{u}$

Possible Answers:

$\vec{v}\cdot\vec{u}=5xe^x+32+2x+6x^3$

$\vec{v}\cdot\vec{u}=x^2+2xe^x+4x+32$

$\vec{v}\cdot\vec{u}=2x^2+e^x+5x+32$

$\vec{v}\cdot\vec{u}=2x^2+7x+32$

Correct answer:

$\vec{v}\cdot\vec{u}=5xe^x+32+2x+6x^3$

Explanation:

Calculate the dot product of the 2 vectors.

In general,

$\vec{v} = (v_1,v_2,v_3)$

$\vec{u} = (u_1,u_2,u_3)$

$\vec{v} \cdot \vec{u} = v_1 * u_1 + v_2 * u_2 + v_3 * u_3$

Solution:

$\vec{v}\cdot\vec{u}=(5x,16+x,2x^2)\cdot(e^x,2,3x)$

$\vec{v}\cdot\vec{u}=5x*e^x+(16+x)*2+2x^2*3x$

$\vec{v}\cdot\vec{u}=5xe^x+32+2x+6x^3$

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Calculus 2 : Parametric, Polar, and Vector

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #254 : Vector

Example Question #255 : Vector

Example Question #256 : Vector

Example Question #4 : Vector Form

Example Question #257 : Vector

Example Question #101 : Vector Calculations

Example Question #102 : Vector Calculations

Example Question #103 : Vector Calculations

Example Question #104 : Vector Calculations

Example Question #105 : Vector Calculations

All Calculus 2 Resources

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