Question 4
<h3>Answer: Choice D</h3>
Explanation:
If the initial point is the origin, the coordinates of the terminal point form the vector itself in component form. We go from (-8,6) to <-8,6>. The notation change is from ordered pair to vector format.
We have a right triangle with legs of 8 and 6 units. The pythagorean theorem will help us determine the hypotenuse is 10. Therefore, the vector length is 10 units and we would say ||v|| = 10. We have a 6-8-10 right triangle.
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Question 5
<h3>Answer: Choice B</h3>
Explanation:
Vector v starts at (2,5) and ends at (-3,-2).
The x component of the vector is x2-x1 = -3-2 = -5 meaning we move 5 units to the left when going from the start point to the endpoint.
At the same time we move 7 units down because y2-y1 = -2-5 = -7 which is the y component of the vector.
The component form of vector v is
v = <-5, -7>
it says "move 5 units left, 7 units down".
Apply the pythagorean theorem to find the length of the vector.
a^2+b^2 = c^2
c = sqrt(a^2 + b^2)
||v|| = sqrt( (-5)^2 + (-7)^2 )
||v|| = 8.602 which is approximate.
Now let's use the arctangent function to find the angle
theta = arctan(b/a)
theta = arctan(-7/(-5))
theta = 54.462 which is also approximate.
There's a problem however. This angle is in Q1 but the vector <-5,-7> is in Q3. An easy fix is to add on 180 to rotate to the proper quadrant.
54.462+180 = 234.462
which is the proper approximate angle for theta.
