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 the given vectors, you can use the following steps:
Step 1: Determine the Magnitude
The magnitude of a vector is calculated using the Pythagorean theorem. For a vector (a, b), the magnitude can be found using the formula:
Magnitude = √(a^2 + b^2)
Let's calculate the magnitude for vector (a) = -8i + 6j:
Magnitude = √((-8)^2 + 6^2)
= √(64 + 36)
= √100
= 10
So, the magnitude of vector (a) is 10.
Similarly, let's calculate the magnitude for vector (b) = -5i - 12j:
Magnitude = √((-5)^2 + (-12)^2)
= √(25 + 144)
= √169
= 13
So, the magnitude of vector (b) is 13.
Step 2: Determine the Direction
The direction of a vector can be found using trigonometric functions. We can use the inverse tangent (arctan) function to find the angle.
For a vector (a, b), the direction θ can be found using the formula:
θ = arctan(b/a)
Let's calculate the direction for vector (a) = -8i + 6j:
θ = arctan(6/-8)
≈ -36.87°
So, the direction of vector (a) is approximately -36.87°.
Similarly, let's calculate the direction for vector (b) = -5i - 12j:
θ = arctan(-12/-5)
≈ 66.04°
So, the direction of vector (b) is approximately 66.04°.
Therefore, the magnitude and direction of the given vectors are:
(a) magnitude = 10, direction = -36.87°
(b) magnitude = 13, direction = 66.04°