Write XY(with line over it) as the sum of unit vectors for X(8,2,-9) and Y(-12,-1,10).
There's an example just like this in my book but I really don't understand it.
To write XY with a line over it as the sum of unit vectors for X(8,2,-9) and Y(-12,-1,10), we need to find the unit vectors corresponding to each of the given vectors and then express XY with a line over it as the sum of these unit vectors.
Step 1: Find the magnitude of vector X and vector Y.
- The magnitude of vector X, denoted as ||X||, can be calculated using the formula: ||X|| = sqrt(X1^2 + X2^2 + X3^2), where X1, X2, and X3 are the components of vector X.
- For X(8,2,-9), ||X|| = sqrt(8^2 + 2^2 + (-9)^2) = sqrt(64 + 4 + 81) = sqrt(149).
- Similarly, we can find the magnitude of vector Y, denoted as ||Y||, using the same formula:
- For Y(-12,-1,10), ||Y|| = sqrt((-12)^2 + (-1)^2 + 10^2) = sqrt(144 + 1 + 100) = sqrt(245).
Step 2: Find the unit vectors for X and Y.
- The unit vector for a given vector V is calculated by dividing each component of V by its magnitude (||V||). It is denoted as V̂.
- For vector X, the unit vector X̂ is calculated as follows:
- X̂ = (X1/||X||, X2/||X||, X3/||X||) = (8/sqrt(149), 2/sqrt(149), -9/sqrt(149)).
- Similarly, we can find the unit vector for vector Y, denoted as Y^:
- Ŷ = (Y1/||Y||, Y2/||Y||, Y3/||Y||) = (-12/sqrt(245), -1/sqrt(245), 10/sqrt(245)).
Step 3: Express XY with a line over it as the sum of these unit vectors.
- Now, we can express XY with a line over it (denoted as XY̅) as the sum of the unit vectors X̂ and Ŷ:
- XY̅ = X̂ + Ŷ = (8/sqrt(149), 2/sqrt(149), -9/sqrt(149)) + (-12/sqrt(245), -1/sqrt(245), 10/sqrt(245)).
This gives us the desired expression of XY̅ as the sum of unit vectors for X(8,2,-9) and Y(-12,-1,10).