Final answer:
To find the projection of vector u=[1,-2,3] on the vector v=[1,-1,0], first compute the unit vector of v, then use the dot product of u and v, and multiply the result with the unit vector of v to get the projection.
Step-by-step explanation:
The projection of vector u onto vector v is found by using the dot product and scalar multiplication of the unit vector in the direction of v.
Firstly, we normalize vector v to find its unit vector î. We do this by dividing v by its magnitude:
v = [1, -1, 0] with magnitude |v| = √(1^2 + (-1)^2 + 0^2) = √2.
Unit vector î = [1/√2, -1/√2, 0].
Next, we use the dot product of u and v:
u · v = (1)(1) + (-2)(-1) + (3)(0) = 1 + 2 + 0 = 3.
The projection of u onto v, which we can call projvu, is then the product of this dot product and the unit vector î:
projvu = (3)[1/√2, -1/√2, 0] = [<3/√2>, <-3/√2>, 0].
So, the projection of vector u on vector v is [<3/√2>, <-3/√2>, 0].