A
Take the product of corresponding components, then take the total:
u • v = (9i - 6j) • (-3i - 2j) = 9 (-3) + (-6) (-2) = -15
u • w = (9i - 6j) • (19i + 15j) = 9 × 19 + (-6) 15 = 81
Alternatively, use the definition and algebraic properties of the dot product.
• For any vector x,
x • x = ||x||²
where ||x|| is the magnitude of x.
• i • i = j • j = k • k = 1
• i • j = j • k = k • i = 0
• For any scalar c and any two vectors x and y,
cx • y = x • cy = c (x • y)
• For any vectors x, y, and z,
(x + y) • z = z • (x + y) = x • z + y • z
Then the products you want are
u • v = (9i - 6j) • (-3i - 2j)
… = -27 (i • i) + 18 (j • i) - 18 (i • j) + 12 (j • j)
… = -27 + 12 = -15
u • w = (9i - 6j) • (19i + 15j)
… = 171 (i • i) - 114 (j • i) + 135 (i • j) - 90 (j • j)
… = 171 - 90 = 81
