Write an expression for a unit vector at 45∘ clockwise from the x-axis.
Express your answer in terms of the unit vectors i^ and j^.
To find the expression for a unit vector at a specific angle from the x-axis, we can use the concept of vector components.
Let's assume the angle between the x-axis and the desired vector is 45 degrees in the clockwise direction.
First, we'll determine the x-component and y-component of the vector using trigonometry.
The x-component is found by multiplying the magnitude of the vector (which is 1 since we want a unit vector) by the cosine of the angle. In this case, the angle is 45 degrees, so the x-component is: cos(45°) = √2/2.
The y-component is found by multiplying the magnitude of the vector by the sine of the angle. For an angle of 45 degrees, the y-component is: sin(45°) = √2/2.
Therefore, the expression for the unit vector at 45 degrees clockwise from the x-axis can be written as: (√2/2)i^ + (√2/2)j^.
To find the unit vector at 45° clockwise from the x-axis, we can start by considering the unit vector along the x-axis, which is written as i^.
To rotate this vector 45° clockwise, we can use the rotation matrix for a 2D coordinate system, which is given by:
cos(θ) -sin(θ)
sin(θ) cos(θ)
For a clockwise rotation of 45°, we have θ = -45°. Plugging this into the rotation matrix, we get:
cos(-45°) -sin(-45°)
sin(-45°) cos(-45°)
Now, let's evaluate these trigonometric values:
cos(-45°) = cos(45°) = √2/2
sin(-45°) = -sin(45°) = -√2/2
Substituting these values into the rotation matrix, we have:
√2/2 -(-√2/2)
-√2/2 √2/2
Simplifying this, we find the resulting matrix as:
√2/2 √2/2
-√2/2 √2/2
The unit vector at 45° clockwise from the x-axis can now be expressed as the vector (a, b) multiplied by the unit vector i^:
(a*i^) + (b*j^)
Here, a = √2/2 and b = √2/2, so the final expression for the unit vector at 45° clockwise from the x-axis is:
(√2/2 * i^) + (√2/2 * j^)
.707i-.707j
assuming +j is up the y axis...it doesnt have to be.