A vector with magnitude 9 points in a direction 190 degrees counterclockwise from the positive x axis. Write in component form

Step-by-step explanation:

Component form of a vector is given by , where represents change in x-value and represents change in y-value. The magnitude of a vector is correlated the Pythagorean Theorem. For vector , the magnitude is .

190 degrees counterclockwise from the positive x-axis is 10 degrees below the negative x-axis. We can then draw a right triangle 10 degrees below the horizontal with one leg being , one leg being , and the hypotenuse of the triangle being the magnitude of the vector, which is given as 9.

In any right triangle, the sine/sin of an angle is equal to its opposite side divided by the hypotenuse, or longest side, of the triangle.

Therefore, we have:

To find the other leg, , we can also use basic trigonometry for a right triangle. In right triangles only, the cosine/cos of an angle is equal to its adjacent side divided by the hypotenuse of the triangle. We get:

Verify that

Therefore, the component form of this vector is