Final answer:
The component form of vector AB is <-7, -5>. Its magnitude is approximately 8.60 units, and the direction angle is approximately 215.5°.
Step-by-step explanation:
To find the component form of the vector (AB) with initial point A(1,-3) and terminal point B(-6,-8), you subtract the coordinates of A from the coordinates of B. The results are as follows:
- x-component of AB: -6 - 1 = -7
- y-component of AB: -8 - (-3) = -5
The component form of vector AB is thus <-7, -5>.
To calculate the magnitude of vector AB, you use the Pythagorean theorem:
√((-7)^2 + (-5)^2) = √(49 + 25) = √74 ≈ 8.60 units
For the direction angle of vector AB, known as θ, you should use the inverse tangent function:
θ = arctan(y-component/x-component) = arctan(-5/-7) ≈ 35.5° (However, since the vector is in the third quadrant, you must add 180° to find the standard position angle, which makes it 35.5° + 180° = 215.5°)
The magnitude of vector AB is approximately 8.60 units, and its direction angle in standard position is approximately 215.5°.