Gaurav P. answered • 08/10/20
Computer Science Student with Teaching Experience
Your question is unclear as to which product you're looking for, the dot product or the cross product, so I'll answer for both.
The i and j notations for vectors refer to their coordinates, e.g. a vector that is 2i+3j is one that moves 2 units in the x-direction and 3 units in the y-direction. This can also be notated as (2, 3), and that's the notation I'll be using in this answer to keep things simple.
In order to get either product, we have to do the subtraction in parentheses first:3v - 4w = 3 * (2, -2) - 4 * (0, -2). When multiplying a vector by a scalar number, you just multiply the coordinates by the scalar, so this becomes (6, -6) - (0, -8). Performing this subtraction by subtracting corresponding coordinates, we get 3v - 4w) = (6, 2). Also, 5u = 5 * (-3, 2) = (-15, 10).
Now, the dot product of two vectors is a scalar quantity. The dot product of the two vectors (-15, 10) • (6, 2) is the sum of the products of the corresponding coordinates. This is -15 * 6 + 10 * 2, which is -90 + 20, and that is -70.
The cross product of two vectors, on the other hand, is a vector quantity, and to get it is a bit more complicated. The vector is at right angles to both vectors being multiplied, so this means that we have to branch out from 2D space into 3D space: our product will thus be (-15, 10, 0) × (6, 2, 0). (We can expand it like this since if you think of the vectors in i, j, k notation, there is an implicit +0k in each one, and k represents z-coordinates.) The formula goes like this: (a, b, c) × (d, e, f) = (bf - ce, cd - af, ae - bd). (If you're familiar with matrices, this is the determinant of the 3×3 matrix [[i, j, k], [a, b, c], [d, e, f]].) Substituting the numbers a = -15, b = 10, c = 0, d = 6, e = 2, and f = 0, we get (0 - 0, 0 - 0, -30 - 60), or (0, 0, -90), or -90k for the cross product.
