Iam stuck with this question including negative sign. Please I need help. Determine the magnitude and direction of this vectors (a) -8i + 6j (b) -5i - 12j

To determine the magnitude and direction of a vector, we can follow these steps:

Step 1: Identify the components of the vector.
(a) For vector (a), the components are -8i (representing the x-component) and 6j (representing the y-component).
(b) For vector (b), the components are -5i (representing the x-component) and -12j (representing the y-component).

Step 2: Calculate the magnitude.
The magnitude of a vector can be calculated using the Pythagorean theorem. For a vector with components (a, b), the magnitude ||v|| is given by the formula:

||v|| = sqrt(a² + b²)

For vector (a):
||a|| = sqrt((-8)² + 6²) = sqrt(64 + 36) = sqrt(100) = 10

For vector (b):
||b|| = sqrt((-5)² + (-12)²) = sqrt(25 + 144) = sqrt(169) = 13

So, the magnitude of vector (a) is 10 and the magnitude of vector (b) is 13.

Step 3: Determine the direction.
The direction of a vector can be found by calculating the angle it makes with the positive x-axis (usually measured counterclockwise).

For vector (a):
To find the direction of vector (a), we can use the tangent function.
tan(theta) = (y-component / x-component)

tan(theta) = 6 / (-8) = -3/4

Using an inverse tangent calculator, we find that theta is approximately -36.87 degrees.

However, since the x-component is negative, the angle is in the second quadrant. Hence, the direction of vector (a) is 180° - 36.87° = 143.13° counterclockwise from the positive x-axis.

For vector (b):
tan(theta) = (-12) / (-5) = 12/5

Using an inverse tangent calculator, we find that theta is approximately 67.38 degrees.

Since both the x-component and y-component are negative, the angle is in the third quadrant. Hence, the direction of vector (b) is 180° + 67.38° = 247.38° counterclockwise from the positive x-axis.

So, the magnitude and direction of vector (a) are 10 units and 143.13°, respectively. And the magnitude and direction of vector (b) are 13 units and 247.38°, respectively.