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Let’s investigate a vector, V. It can be written in vector components.
We can break it up into Vx in the i direction and Vy in the j direction.
A vector of length one is called a unit vector:
i with an arrow over it representing the unit vector in the x direction,
and j with an arrow over it, representing the unit vector in the y direction.
The unit vector, i, is multiplied by the scalar, Vx, to indicate the x component of the vector.
And unit vector, j, is multiplied by the scalar, Vy, to indicate the y component of the vector.
So, vector V can be written as vector components.
Let’s give vector V a value of one Newton and an angle from the x-axis of 36.87 degrees.
We can look at its components using trigonometry.
Vx is equal to V cos Θ, which is "1" cosine 36.87 degrees. That equals 0.8 Newtons.
In component form, we can say a one Newton vector has a vector component
equal to 0.8 times the unit vector, i.
We can use the same process to find Vy, which is 0.6 Newtons.
Our vector, V, can now be expressed as 0.8 times unit vector i, plus 0.6 times unit vector j.
With the vector components calculated,
we can express the vector direction and magnitude in vector notation.
You can also find the angle and magnitude when given vector components.
Here, vector, V, equals 25 i plus 17 j pounds (lbs).
This means, the component in the direction of unit vector i, the x-axis, is 25 pounds (lbs).
And the one in the direction of unit vector j, on the y-axis, is 17 pounds (lbs).
The magnitude of V is 30.2 pounds (lbs), acting at an angle of 30.23 degrees from the x-axis.
For more details on the use of the Pythagorean Theorem and trigonometry to solve this vector,
refer to the 2D Scalar Components Video.