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

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Question

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.

Answer

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

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

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

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

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

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

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

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Question

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.

Answer

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

$c^2=11^2+21^2-2\cdot11\cdot21\cos(21^{\circ})$

$c^2=121+441-462\cos(21^{\circ})$

$c^2=562-462\cos(21^{\circ})$

$c^2=562-462\cdot0.9336$

$\sqrt{c^2}=\sqrt{130.6858}$

$c=\sqrt{130.6858}\approx11.43$

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Question

If $\left | v\right |=8$ , and $\left | w\right |=22$ , and the angle in between them is $103^{\circ}$ , find the magnitude of the resultant vector.

Answer

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 $C=103^{\circ}$ , then

$c^2=8^2+22^2-2\cdot8\cdot22\cos(103^{\circ})$

$c^2=64+484-16\cdot22\cos(103^{\circ})$

$c^2=548-352\cos(103^{\circ})$

$c^2=548-352\cdot-0.2249511$

$\sqrt{c^2}=\sqrt{627.1828}$

$c=\sqrt{627.1828}\approx25.04$

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Question

If $\left | v\right |=1$ , and $\left | w\right |=38$ , and the angle in between them is $116^{\circ}$ , find the magnitude of the resultant vector.

Answer

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 $C=116^{\circ}$ , then

$c^2=1^2+38^2-2\cdot1\cdot38\cos(116^{\circ})$

$c^2=1+1444-2\cdot38\cos(116^{\circ})$

$c^2=1445-76\cos(116^{\circ})$

$c^2=1445-76\cdot-.4383711$

$\sqrt{c^2}=\sqrt{1478.316}$

$c=\sqrt{1478.316}\approx38.45$

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Question

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.

Answer

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

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

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

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

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

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

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Question

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.

Answer

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

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

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

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

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

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

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

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Question

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.

Answer

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

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

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

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

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

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

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Question

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.

Answer

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

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

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

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

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

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

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

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Question

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.

Answer

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

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

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

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

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

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

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

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Question

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.

Answer

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

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

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

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

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

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

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

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Question

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.

Answer

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

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

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

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

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

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

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

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Question

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.

Answer

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

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

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

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

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

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

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

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Question

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.

Answer

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.

Compare your answer with the correct one above

Question

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.

Answer

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

$c^2=11^2+21^2-2\cdot11\cdot21\cos(21^{\circ})$

$c^2=121+441-462\cos(21^{\circ})$

$c^2=562-462\cos(21^{\circ})$

$c^2=562-462\cdot0.9336$

$\sqrt{c^2}=\sqrt{130.6858}$

$c=\sqrt{130.6858}\approx11.43$

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Question

If $\left | v\right |=8$ , and $\left | w\right |=22$ , and the angle in between them is $103^{\circ}$ , find the magnitude of the resultant vector.

Answer

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 $C=103^{\circ}$ , then

$c^2=8^2+22^2-2\cdot8\cdot22\cos(103^{\circ})$

$c^2=64+484-16\cdot22\cos(103^{\circ})$

$c^2=548-352\cos(103^{\circ})$

$c^2=548-352\cdot-0.2249511$

$\sqrt{c^2}=\sqrt{627.1828}$

$c=\sqrt{627.1828}\approx25.04$

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Question

If $\left | v\right |=1$ , and $\left | w\right |=38$ , and the angle in between them is $116^{\circ}$ , find the magnitude of the resultant vector.

Answer

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 $C=116^{\circ}$ , then

$c^2=1^2+38^2-2\cdot1\cdot38\cos(116^{\circ})$

$c^2=1+1444-2\cdot38\cos(116^{\circ})$

$c^2=1445-76\cos(116^{\circ})$

$c^2=1445-76\cdot-.4383711$

$\sqrt{c^2}=\sqrt{1478.316}$

$c=\sqrt{1478.316}\approx38.45$

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Question

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.

Answer

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

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

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

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

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

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

Compare your answer with the correct one above

Question

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.

Answer

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

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

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

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

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

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

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

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Question

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.

Answer

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

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

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

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

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

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

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Question

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.

Answer

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

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

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

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

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

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

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

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