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^.
1/√2i - 1/√2j
To determine the expression for a unit vector at a given angle, we can use the concept of a vector's components.
Let's break down the problem. We want a unit vector at an angle of 45∘ clockwise from the x-axis.
The unit vector in the x-direction is represented by i^, and the unit vector in the y-direction is represented by j^. Since we need to find a vector that makes a 45∘ angle with the x-axis, we can use the trigonometric properties of a right triangle.
In a right triangle, the cosine of an angle is equal to the adjacent side divided by the hypotenuse, while the sine of an angle is equal to the opposite side divided by the hypotenuse.
Since we want a unit vector, the magnitude of the vector will be 1.
Now, let's construct a right triangle with an angle of 45∘. In this triangle, the adjacent side is the x-component of the vector, the opposite side is the y-component of the vector, and the hypotenuse is 1.
The cosine of 45∘ is (√2)/2, and the sine of 45∘ is (√2)/2.
Using this information, we can write the expression for the unit vector as follows:
cos(45∘) * i^ + sin(45∘) * j^
Substituting the values of cosine and sine, the expression simplifies to:
(√2)/2 * i^ + (√2)/2 * j^
Therefore, the expression for the unit vector at 45∘ clockwise from the x-axis is:
(√2)/2 * i^ + (√2)/2 * j^