Let u have a length of 20 and direction of 60 degrees, and v= 5i-3j
what is u?
what is abs(2v-u)?
To determine the vector u, which has a length of 20 and a direction of 60 degrees, follow these steps:
1. Start with the given information about the length and direction of the vector u.
2. Recall that a 2D vector can be represented as u = xi + yj, where i and j are the unit vectors along the x and y axes, respectively.
3. Use the formula for vector magnitude to find the values of x and y. Since the length of u is given as 20, we have:
magnitude(u) = √(x^2 + y^2) = 20
From this equation, we can find the relationship between x and y.
4. The direction of the vector u is given as 60 degrees. Notice that the direction angle refers to the angle between the positive x-axis and the vector u. To convert this to standard Cartesian coordinates, subtract 90 degrees from the given value, i.e., 60 - 90 = -30 degrees (angle measured clockwise from the positive x-axis).
5. Use trigonometric functions to find the values of x and y. The trigonometric function of cosine (cos) gives the value of x, and the function of sine (sin) gives the value of y. Since the direction angle is -30 degrees, we have:
x = cos(-30°)
y = sin(-30°)
6. Substitute the values of x and y into the equation from step 3 to solve for x and y.
√(x^2 + y^2) = 20
Plug in the values from step 5:
√[cos^2(-30°) + sin^2(-30°)] = 20
Simplify the equation and solve for x and y.
Once you have the values of x and y, you will have the vector u in the form u = xi + yj.
To calculate abs(2v - u):
1. Begin with the given vectors v and u.
2. Multiply the vector v by 2. Since v = 5i - 3j, multiplying it by 2 gives us:
2v = 2(5i - 3j)
3. Subtract vector u from the resulting vector 2v:
2v - u = (2(5i - 3j)) - (xi + yj)
4. Distribute the multiplication and simplify:
2v - u = (10i - 6j) - (xi + yj)
5. Combine like terms:
2v - u = (10 - x)i + (-6 - y)j
6. The resulting vector 2v - u can be expressed as (10 - x)i + (-6 - y)j.