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Calculus 2 : Parametric, Polar, and Vector

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Example Question #1124 : Calculus Ii

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

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

Possible Answers:

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

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

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

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

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 #101 : Vector Calculations

Find the sum of the vectors and if

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

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

Possible Answers:

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

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

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

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

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 : Parametric, Polar, and Vector

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #1124 : Calculus Ii

Example Question #101 : Vector Calculations

All Calculus 2 Resources

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