Question

Vectors x = [15, 20, 18, 12]

[18]
(7)
Y = (2) and Z = (11)
(15) (5)
(20) (10)
Determine:
a. Y + Z
b. Y - Z
c. x(Y + Z)
d. XY + XZ"

Answer 1

The question requires understanding vector operations such as addition, subtraction, and dot product, each involving algebraic manipulation of vector components in a step-by-step process resulting in the computation of the vector sums, differences, and products.

Step-by-step explanation:

The student's question involves vector arithmetic and vector scalar products in high school-level mathematics.

To perform the calculations, here are the steps:

Addition of vectors Y and Z (Y + Z): Add corresponding components of vectors Y and Z.
[18 + 7, 2 + 11, 5 + 15, 10 + 20] = [25, 13, 20, 30]

Subtraction of vectors Y and Z (Y - Z): Subtract corresponding components of vector Z from vector Y.
[18 - 7, 2 - 11, 5 - 15, 10 - 20] = [11, -9, -10, -10]

Dot product of vector x and (Y + Z): Multiply each component of vector x by the corresponding component of the result from step 1, and then add all the products together.
15×25 + 20×13 + 18×20 + 12×30 = 375 + 260 + 360 + 360

= 1355

Dot products XY and XZ and their addition (XY + XZ): Compute the dot products separately and then add them.
(15×18 + 20×2 + 18×5 + 12×10) + (15×7 + 20×11 + 18×15 + 12×20) = (270 + 40 + 90 + 120) + (105 + 220 + 270 + 240)

= 520 + 835

= 1355