Use the given vector to find a vector b such that compab = 4.
a = ‹4, 0, 1›
‹s, t, >
To find a vector b such that the scalar projection of a onto b (compab) is 4, we need to determine the values of vector b, which can be represented as ‹s, t, ›.
First, we need to find the formula for the scalar projection of vector a onto vector b. The scalar projection of vector a onto vector b (compab) is given by:
compab = (a . b) / ||b||
Where 'a . b' represents the dot product of vectors a and b, and '||b||' represents the magnitude (length) of vector b.
Now, let's calculate each part of the formula step by step:
1. Calculate 'a . b':
To find 'a . b', we need to take the dot product of vectors a and b. However, since we don't have the values of vector b, we will use the variables 's' and 't' to represent them. So, substitute the values of vector a into the dot product formula:
a . b = (4 * s) + (0 * t) + (1 * )
2. Calculate ||b||:
To find ||b||, we need to calculate the magnitude (length) of vector b. However, since we don't know the values of 's' and 't' yet, we can represent ||b|| as √(s^2 + t^2).
3. Solve for 's' and 't':
Now, substitute the calculated values of 'a . b' and ||b|| into the original formula for the scalar projection:
4 = [(4 * s) + (0 * t) + (1 * )] / √(s^2 + t^2)
Square both sides of the equation to eliminate the square root:
16 = [(4 * s) + (0 * t) + (1 * )]^2 / (s^2 + t^2)
Expand the numerator:
16 = [16 * s^2 + 2 * s * t + 1 * ^2] / (s^2 + t^2)
Multiply both sides by (s^2 + t^2):
16(s^2 + t^2) = 16s^2 + 2st + 1
Simplify the equation:
16s^2 + 16t^2 = 16s^2 + 2st + 1
Combine like terms:
16t^2 = 2st + 1
Now, you can solve this equation for 't' in terms of 's':
16t^2 - 2st - 1 = 0
Using the quadratic formula, solve for 't':
t = (-(-2s) ± √((-2s)^2 - 4 * 16 * -1)) / (2 * 16)
Simplify the expression under the square root:
t = (2s ± √(4s^2 + 64)) / 32
Therefore, the vector b can be represented as ‹s, (2s ± √(4s^2 + 64)) / 32, ›.
what is "compab" ?
What are s and t?
What does <s,t> mean?
Is a the three-dimensional vector
‹4, 0, 1› ?
There must be more backgeound information to your question.