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Calculus 2 : Vector

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Example Questions

Example Question #101 : 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 #102 : 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:

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

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

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

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

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 #103 : 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:

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

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

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

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

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 #104 : 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:

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

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

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

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

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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Example Question #105 : Vector Calculations

$\newline\vec{v} =\left(\left(\frac{\sqrt{2}+1}{2}\right),\frac{2x}{e^{x}},-2\right)\newline\vec{u} =\left(\left(4+x\right),ln(x),2\right)\newline$

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

Possible Answers:

$\vec{v}\cdot\vec{u}=\frac{2xln(x)}{e^x}$

$\vec{v}\cdot\vec{u}=2\sqrt{2}+2+\frac{2xln(x)}{e^x}$

$\vec{v}\cdot\vec{u}=2\sqrt{2}+4+\frac{2xln(x)}{e^x}$

$\vec{v}\cdot\vec{u}=2\sqrt{2}-2+\frac{(\sqrt2+1)}{2}x+\frac{2xln(x)}{e^x}$

Correct answer:

$\vec{v}\cdot\vec{u}=2\sqrt{2}-2+\frac{(\sqrt2+1)}{2}x+\frac{2xln(x)}{e^x}$

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:

$\newline\vec{v} =\left(\left(\frac{\sqrt{2}+1}{2}\right),\frac{2x}{e^{x}},-2\right)\newline\vec{u} =\left(\left(4+x\right),ln(x),2\right)\newline$

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

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

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

$\vec{v}\cdot\vec{u}=2\sqrt{2}-2+\frac{(\sqrt2+1)}{2}x+\frac{2xln(x)}{e^x}$

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Example Question #106 : Vector Calculations

Find the sum of the vectors and if

$u=\left<3, 7\right>$

$v=\left<7, 10\right>$ .

Possible Answers:

$u+v=\left<17,38\right>$

$u+v=\left<21,70\right>$

$u+v=\left<10,17\right>$

$u+v=\left<4,3\right>$

Correct answer:

$u+v=\left<10,17\right>$

Explanation:

For the two vectors

$a=\left<a_1,a_2\right>$ and $b=\left<b_1,b_2\right>$

The sum of the two vectors a+b is

$\left<a_1, a_2\right> + \left<b_1, b_2\right>= \left<a_1+b_1, a_2+b_2\right>$

For the vectors u and v in this problem the sum u+v is

$\left<3, 7\right> + \left<7, 10\right>$

$= \left<3+7, 7+10\right>$

$=\left <10, 17\right>$

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Calculus 2 : Vector

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

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

Example Question #106 : Vector Calculations

All Calculus 2 Resources

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