For the vectors x=(0,6,-5), y=(2,1,4) and z=(-2,3,-1) what is the value of (2x+y)·(3y-z)? (dot product question and each letter is a vector)
SOLVED
good, that is a long one :)
add, then multiply, then add
To find the value of (2x+y)·(3y-z), we first need to find the dot product of (2x+y) and (3y-z).
Let's calculate the expression step-by-step:
Step 1: Determine the value of 2x+y
2x means multiplying each component of the vector x by 2, and y stays the same.
2x = 2 * (0, 6, -5) = (0*2, 6*2, -5*2) = (0, 12, -10)
y = (2, 1, 4)
So, 2x+y = (0, 12, -10) + (2, 1, 4) = (0+2, 12+1, -10+4) = (2, 13, -6)
Step 2: Determine the value of 3y-z
3y means multiplying each component of the vector y by 3, and z is subtracted as it is.
3y = 3 * (2, 1, 4) = (2*3, 1*3, 4*3) = (6, 3, 12)
z = (-2, 3, -1)
So, 3y-z = (6, 3, 12) - (-2, 3, -1) = (6+2, 3-3, 12+1) = (8, 0, 13)
Step 3: Determine the value of (2x+y) · (3y-z)
To find the dot product, we multiply the corresponding components of the two vectors and then sum them up.
(2x+y) · (3y-z) = (2, 13, -6) · (8, 0, 13) = 2*8 + 13*0 + -6*13
= 16 - 78
= -62
Therefore, the value of (2x+y)·(3y-z) is -62.
To find the value of the dot product (2x + y) · (3y - z), we need to perform two steps:
Step 1: Calculate the vector (2x + y) and (3y - z).
Step 2: Calculate the dot product of the calculated vectors from step 1.
Let's go through each step:
Step 1: Calculate the vector (2x + y) and (3y - z).
To find the vector (2x + y), we need to multiply the scalar 2 to each component of vector x and then add it to each corresponding component of vector y.
2x = (2 * 0, 2 * 6, 2 * -5) = (0, 12, -10)
Now add it to vector y:
(2x + y) = (0, 12, -10) + (2, 1, 4) = (0 + 2, 12 + 1, -10 + 4) = (2, 13, -6)
Similarly, let's calculate (3y - z):
3y = (3 * 2, 3 * 1, 3 * 4) = (6, 3, 12)
Now subtract vector z from it:
(3y - z) = (6, 3, 12) - (-2, 3, -1) = (6 + 2, 3 - 3, 12 + 1) = (8, 0, 13)
So (2x + y) = (2, 13, -6) and (3y - z) = (8, 0, 13).
Step 2: Calculate the dot product of the calculated vectors.
To calculate the dot product of two vectors, we need to multiply the corresponding components of the vectors and then sum them up.
(2x + y) · (3y - z) = (2 * 8) + (13 * 0) + (-6 * 13) = 16 + 0 - 78 = -62
Therefore, the value of (2x + y) · (3y - z) is -62.