The vector u has length 15 and points at an angle of 25 degrees
The vector v has length 20 and points at an angle of 55 degrees
Find u*v=?
the angle θ between the vectors is 55-25 = 30°.
u•v = |u|*|v|*cosθ
now just plug in your numbers.
To find the dot product of the vectors u and v, we first need to determine the components of each vector. The components can be found using the given information about the lengths and angles of the vectors.
The component of vector u in the x-direction (u_x) can be calculated using the formula:
u_x = length of u * cos(angle of u)
Substituting the values:
u_x = 15 * cos(25 degrees)
Using a calculator:
u_x ≈ 15 * 0.9063 ≈ 13.5945
Similarly, the component of vector u in the y-direction (u_y) can be calculated using the formula:
u_y = length of u * sin(angle of u)
Substituting the values:
u_y = 15 * sin(25 degrees)
Using a calculator:
u_y ≈ 15 * 0.4226 ≈ 6.3390
So the components of vector u are approximately u_x = 13.5945 in the x-direction and u_y = 6.3390 in the y-direction.
We can follow the same process to find the components of vector v:
v_x = length of v * cos(angle of v)
v_x = 20 * cos(55 degrees)
Using a calculator:
v_x ≈ 20 * 0.5736 ≈ 11.4721
v_y = length of v * sin(angle of v)
v_y = 20 * sin(55 degrees)
Using a calculator:
v_y ≈ 20 * 0.8192 ≈ 16.3840
So the components of vector v are approximately v_x = 11.4721 in the x-direction and v_y = 16.3840 in the y-direction.
The dot product (u*v) of u and v can be calculated using the formula:
u*v = u_x * v_x + u_y * v_y
Substituting the values:
u*v = 13.5945 * 11.4721 + 6.3390 * 16.3840
Using a calculator:
u*v ≈ 155.9582 + 103.8266 ≈ 259.7848
Therefore, u*v is approximately 259.7848.