We are asked to determine the components of a vector that has a magnitude of 26 and a direction of 45 degrees. A diagram of the vector is the following:
We notice that the vector and its components form a right triangle where the magnitude of the vector is the hypotenuse and the y and x components are the sides.
First, we will determine the x-components using the trigonometric function cosine, which is defined as:
[tex]\cos \theta=\frac{adjacent}{hypotenuse}[/tex]Substituting the values:
[tex]\cos 45=\frac{x}{26}[/tex]Multiplying both sides by 26 we get:
[tex]26\cos 45=x[/tex]Solving the operations we get:
[tex]18.4=x[/tex]Therefore, the x-component is 18.4 m.
Now, we will determine the y-component using the function sine, which is defined as:
[tex]\sin \theta=\frac{opposite}{hypotenuse}[/tex]Substituting the values:
[tex]\sin 45=\frac{y}{26}[/tex]Now, we multiply both sides by 26, we get:
[tex]26\sin 45=y[/tex]Now, we solve the operations:
[tex]18.4=y[/tex]Therefore, the y-component is 18.4 m.
