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The diagram shows a vector π that has a magnitude of 24.
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The angle between the vector and the π₯-axis is 43 degrees.
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Give this vector in component form.
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Round all numbers in your answer to the nearest whole number.
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In the diagram, we can see the vector π, which has a magnitude of 24 and makes an angle of 43 degrees with the π₯- or horizontal axis.
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Weβre asked to write π in component form, which means expressing it in the form π is equal to π sub π₯π’ hat plus π sub π¦π£ hat.
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π’ hat and π£ hat are unit vectors, where π’ hat represents one unit in the horizontal direction and π£ hat is one unit in the vertical direction.
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The values that we need to find are π sub π₯, which is the magnitude or size of the horizontal component, and π sub π¦, which is the magnitude or size of the vertical component.
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If we draw these onto our diagram, they make two sides of a right-angled triangle.
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The angle of the triangle that weβre given is 43 degrees.
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And relative to that angle, the side π sub π₯ is the adjacent side, π sub π¦ is the opposite side, and then the magnitude or size of the vector, which weβll call π΄, is the hypotenuse.
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Now weβre going to use trigonometry to solve this, so we need to recall SOHCAHTOA.
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Letβs start by finding π sub π₯, which is the adjacent side.
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We already know the hypotenuse, so this tells us we need to use the cosine of the angle.
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So we need to recall that cos π equals adjacent divided by hypotenuse.
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So this gives us cos π is equal to π sub π₯ divided by π΄.
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And then we can multiply both sides by π΄.
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And that gives us π΄ cos π is equal to π sub π₯.
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Now we can put our numbers in.
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We have π΄ is equal to 24 and π is 43 degrees, which gives us 24 times the cos of 43 degrees.
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Now, if we put this into our calculator making sure itβs in degrees, we find that π sub π₯ is equal to 17.55.
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And weβre asked to give our answer to the nearest whole number, so that becomes 18.
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So π sub π₯ is equal to 18
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Now letβs clear some space so we can work out π sub π¦.
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π sub π¦ is the opposite side of the triangle.
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And again we know the hypotenuse.
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So this time weβre going to work with the sine of the angle.
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So from this we recall that sin of π is equal to the opposite divided by the hypotenuse.
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Therefore, sin of π is equal to π sub π¦ divided by π΄.
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Now we can multiply both sides by π΄.
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And we have π΄ sin π is equal to π sub π¦.
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So substituting in our numbers we have π sub π¦ is equal to 24 times the sin of 43 degrees.
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Therefore, π sub π¦ is equal to 16.37.
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Again, we want this to the nearest whole number.
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So that becomes 16.
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Now that we have both π sub π₯ and π sub π¦, we can write the vector π in component form as π is equal to 18π’ hat plus 16π£ hat.
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And that gives us our final answer.