Magnitude and Direction of a Vector Sum: CCSS.Math.Content.HSN-VM.B.4b

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Common Core: High School - Number and Quantity › Magnitude and Direction of a Vector Sum: CCSS.Math.Content.HSN-VM.B.4b

Questions 1 - 10

If $\left | v\right |=4$ , and $\left | w\right |=23$ , and the angle in between them is $149^{\circ}$ , find the magnitude of the resultant vector.

$\left | v+w\right |=27.23$

$\left | v+w\right |=28.13$

$\left | v+w\right |=26.42$

$\left | v+w\right |=27.76$

$\left | v+w\right |=24.57$

Explanation

In order to find the resultants magnitude, we need to use the law of cosines. The law of cosines is $c^2=a^2+b^2-2ab\cos(C)$ . Suppose that , , and $C=149^{\circ}$ , then

$c^2=4^2+23^2-2\cdot4\cdot23\cos(149^{\circ})$

$c^2=16+529-8\cdot23\cos(149^{\circ})$

$c^2=545-184\cos(149^{\circ})$

$c^2=545-184\cdot-0.8571673$

$\sqrt{c^2}=\sqrt{741.7188}$

$c=\sqrt{741.7188}\approx27.23$

If $\left | v\right |=14$ , and $\left | w\right |=1$ , and the angle in between them is $76^{\circ}$ , find the magnitude of the resultant vector.

$\left | v+w\right |=13.79$

$\left | v+w\right |=14.25$

$\left | v+w\right |=17.94$

$\left | v+w\right |=14.09$

$\left | v+w\right |=13.62$

Explanation

In order to find the resultants magnitude, we need to use the law of cosines. The law of cosines is $c^2=a^2+b^2-2ab\cos(C)$ . Suppose that , , and $C=76^{\circ}$ , then

$c^2=14^2+1^2-2\cdot14\cdot1\cos(76^{\circ})$

$c^2=196+1-28\cdot1\cos(76^{\circ})$

$c^2=197-28\cos(76^{\circ})$

$c^2=197-28\cdot0.2419219$

$\sqrt{c^2}=\sqrt{190.2262}$

$c=\sqrt{190.2262}\approx13.79$

If $\left | v\right |=18$ , and $\left | w\right |=19$ , and the angle in between them is $131^{\circ}$ , find the magnitude of the resultant vector.

$\left | v+w\right |=33.67$

$\left | v+w\right |=31.56$

$\left | v+w\right |=32.89$

$\left | v+w\right |=34.90$

$\left | v+w\right |=37.92$

Explanation

In order to find the resultants magnitude, we need to use the law of cosines. The law of cosines is $c^2=a^2+b^2-2ab\cos(C)$ . Suppose that , , and $C=106^{\circ}$ , then

$c^2=18^2+19^2-2\cdot18\cdot19\cos(131^{\circ})$

$c^2=324+361-36\cdot19\cos(131^{\circ})$

$c^2=685-684\cos(131^{\circ})$

$c^2=685-684\cdot-0.656059$

$\sqrt{c^2}=\sqrt{1133.744}$

$c=\sqrt{1133.744}\approx33.67$

If $\left | v\right |=12$ , and $\left | w\right |=34$ , and the angle in between them is $54^{\circ}$ , find the magnitude of the resultant vector.

$\left | v+w\right |=28.64$

$\left | v+w\right |=28.14$

$\left | v+w\right |=29.67$

$\left | v+w\right |=24.87$

$\left | v+w\right |=26.67$

Explanation

In order to find the resultants magnitude, we need to use the law of cosines. The law of cosines is $c^2=a^2+b^2-2ab\cos(C)$ . Suppose that , , and $C=54^{\circ}$ , then

$c^2=12^2+34^2-2\cdot12\cdot34\cos(54^{\circ})$

$c^2=144+1156-24\cdot34\cos(54^{\circ})$

$c^2=1300-816\cos(54^{\circ})$

$c^2=1300-816\cdot0.5877853$

$\sqrt{c^2}=\sqrt{820.3672}$

$c=\sqrt{820.3672}\approx28.64$

If $\left | v\right |=9$ , and $\left | w\right |=28$ , and the angle in between them is $150^{\circ}$ , find the magnitude of the resultant vector.

$\left | v+w\right |=36.08$

$\left | v+w\right |=36.89$

$\left | v+w\right |=35.76$

$\left | v+w\right |=33.98$

$\left | v+w\right |=29.53$

Explanation

In order to find the resultants magnitude, we need to use the law of cosines. The law of cosines is $c^2=a^2+b^2-2ab\cos(C)$ . Suppose that , , and $C=150^{\circ}$ , then

$c^2=9^2+28^2-2\cdot9\cdot28\cos(150^{\circ})$

$c^2=81+784-18\cdot28\cos(150^{\circ})$

$c^2=865-504\cos(150^{\circ})$

$c^2=865-504\cdot-0.8660254$

$\sqrt{c^2}=\sqrt{1301.477}$

$c=\sqrt{1301.477}\approx36.08$

If $\left | v\right |=10$ , and $\left | w\right |=5$ , and the angle in between them is $60^{\circ}$ , find the magnitude of the resultant vector.

$\left | v+w\right |=\sqrt{75}$

$\left | v+w\right |=15$

$\left | v+w\right |=7$

$\left | v+w\right |=\sqrt{65}$

$\left | v+w\right |=\sqrt{85}$

Explanation

In order to find the resultants magnitude, we need to use the law of cosines. The law of cosines is $c^2=a^2+b^2-2ab\cos(C)$ . Suppose that , , and $C=60^{\circ}$ , then

$c^2=10^2+5^2-2\cdot10\cdot5\cos(60^{\circ})$

$c^2=100+25-20\cdot5\cos(60^{\circ})$

$c^2=125-100\cos(60^{\circ})$

$c^2=125-100\cdot\frac{1}{2}$

$\sqrt{c^2}=\sqrt{75}$

$c=\sqrt{75}\approx8.66$

Here is a visual representation of what we just found.

Locs

If $\left | v\right |=16$ , and $\left | w\right |=19$ , and the angle in between them is $150^{\circ}$ , find the magnitude of the resultant vector.

$\left | v+w\right |=33.82$

$\left | v+w\right |=40.58$

$\left | v+w\right |=36.87$

$\left | v+w\right |=31.42$

$\left | v+w\right |=30.8$

Explanation

In order to find the resultants magnitude, we need to use the law of cosines. The law of cosines is $c^2=a^2+b^2-2ab\cos(C)$ . Suppose that , , and $C=150^{\circ}$ , then

$c^2=16^2+19^2-2\cdot16\cdot19\cos(150^{\circ})$

$c^2=256+361-32\cdot19\cos(150^{\circ})$

$c^2=617-608\cos(150^{\circ})$

$c^2=617-608\cdot-0.8660254$

$\sqrt{c^2}=\sqrt{1143.543}$

$c=\sqrt{1143.543}\approx33.82$

If $\left | v\right |=34$ , and $\left | w\right |=17$ , and the angle in between them is $10^{\circ}$ , find the magnitude of the resultant vector.

$\left | v+w\right |=17.51$

$\left | v+w\right |=17.84$

$\left | v+w\right |=18.09$

$\left | v+w\right |=14.69$

$\left | v+w\right |=13.56$

Explanation

In order to find the resultants magnitude, we need to use the law of cosines. The law of cosines is $c^2=a^2+b^2-2ab\cos(C)$ . Suppose that , , and $C=10^{\circ}$ , then

$c^2=17^2+34^2-2\cdot34\cdot17\cos(10^{\circ})$

$c^2=289+1156-34\cdot34\cos(10^{\circ})$

$c^2=1445-1156\cos(10^{\circ})$

$\sqrt{c^2}=\sqrt{306.562}$

$c=\sqrt{306.562}\approx17.51$

If $\left | v\right |=18$ , and $\left | w\right |=16$ , and the angle in between them is $26^{\circ}$ , find the magnitude of the resultant vector.

$\left | v+w\right |=7.89$

$\left | v+w\right |=11.43$

$\left | v+w\right |=4.67$

$\left | v+w\right |=9.62$

$\left | v+w\right |=20.27$

Explanation

In order to find the resultants magnitude, we need to use the law of cosines. The law of cosines is $c^2=a^2+b^2-2ab\cos(C)$ . Suppose that , , and $C=60^{\circ}$ , then

$c^2=18^2+16^2-2\cdot18\cdot16\cos(26^{\circ})$

$c^2=324+256-36\cdot16\cos(26^{\circ})$

$c^2=580-576\cos(26^{\circ})$

$\sqrt{c^2}=\sqrt{62.2946}$

$c=\sqrt{62.2946}\approx7.89$

If $\left | v\right |=11$ , and $\left | w\right |=21$ , and the angle in between them is $21^{\circ}$ , find the magnitude of the resultant vector.

$\left | v+w\right |=11.43$

$\left | v+w\right |=12$

$\left | v+w\right |=11.5$

$\left | v+w\right |=11.6$

$\left | v+w\right |=\sqrt{85}$

Explanation

In order to find the resultants magnitude, we need to use the law of cosines. The law of cosines is $c^2=a^2+b^2-2ab\cos(C)$ . Suppose that , , and $C=21^{\circ}$ , then