Video: Finding the Horizontal Component of a Vector

The diagram shows a vector ๐€ that has a magnitude of 22. The angle between the vector and the ๐‘ฅ-axis is 36ยฐ. Work out the horizontal component of the vector. Give your answer to 2 significant figures.

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Video Transcript

The diagram shows a vector ๐€ that has a magnitude of 22. The angle between the vector and the ๐‘ฅ-axis is 36 degrees. Work out the horizontal component of the vector. Give your answer to two significant figures.

In the problem, weโ€™re being asked to solve for the ๐‘ฅ-component of our vector, or ๐‘Ž sub ๐‘ฅ. With the components drawn in on our diagram, we can see that the triangle is a right triangle. And therefore, we can use trigonometry to solve for our unknown side. Letโ€™s recall that we can use SOH-CAH-TOA. SOH, the sin of the angle is equal to the opposite side of the triangle divided by the hypotenuse. CAH, the cos of the angle is equal to the adjacent side of the triangle divided by the hypotenuse. And TOA, the tan of the angle is equal to the opposite side of the triangle divided by the adjacent side of the triangle.

Looking back at our diagram, we are solving for the ๐‘ฅ-component of our vector, weโ€™re given the angle ๐œƒ, and weโ€™re given the length of the hypotenuse. Which one of our trigonometric functions would we choose, sine, cosine, or tangent? The best one to choose is cosine because our vector component is adjacent to the angle. Now, we need to isolate the adjacent side of the triangle, as thatโ€™s what weโ€™re solving for. To do this, we multiply both sides by the hypotenuse. This will cancel out the hypotenuse on the right side of the equation and leave the left side of the equation as the hypotenuse times the cos of ๐œƒ.

We can now use the values from the problem to substitute in for our variables. The hypotenuse had a magnitude of 22, ๐œƒ was 36 degrees, and ๐‘Ž sub ๐‘ฅ represents the horizontal component of our vector. When we multiply 22 by the cos of 36 degrees, we get 17.8. In the problem, we were instructed to give our answers to two significant figures. Looking at our answer right now, itโ€™s given to three significant figures. Therefore, we can round up 17.8 to 18. The horizontal component of a vector that has a magnitude of 22 and makes an angle with the ๐‘ฅ-axis of 36 degrees is 18.

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