A · B = (5)(-5) + (12)(12) = -25 + 144 = 119. Since A and B are in opposite directions, the angle is 180 degrees.

A + B = (5 - 5)i + (12 + 12)j = 0i + 24j.

cos(θ) = (A · B) / (|A||B|) = (5*12 + 12*(-5)) / (√(169) * √(169)) = 0, θ = 90 degrees.

A × B = |i  j  k| |5  12  0| |12 -5  0| = (0 - 0)i - (0 - 0)j + (5*-5 - 12*12)k = -85k.

A · B = 5*5 + 12*(-12) = 25 - 144 = -119. Since A · B < 0, angle is 180 degrees.

cos(θ) = (A · B) / (|A||B|) = (25 - 25) / (√(50) * √(50)) = 0, θ = 90 degrees.

cos(θ) = (A · B) / (|A||B|) = (25 - 25) / (5√2 * 5√2) = 0, θ = 90 degrees.

Magnitude |A| = √(6^2 + 8^2) = 10. Unit vector = (1/10)(6i + 8j) = (3/5)i + (4/5)j.

The y-component of G is -3.

tan(θ) = 12/5; θ = tan^(-1)(12/5) which is approximately 67.38 degrees, closest to 60 degrees.

K + L = (2i + 2j) + (-i + 3j) = (2 - 1)i + (2 + 3)j = 1i + 5j = i + 5j.

A · B = x*1 + 2*y + 3*4 = 0. Thus, x + 2y + 12 = 0. Solving gives x + y = -6.

A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.

A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.

If A and B are perpendicular, then by definition A · B = |A||B|cos(90°) = 0.

The arrangement does not provide enough information to determine who is opposite X.

Since x is between 3 and 7, x - 1 will be between 2 and 6, thus x - 1 < 5 is true.

Since Y = Z and X > Y, it follows that X > Z.

Since Y = Z, and X < Y, it follows that X < Z.

To find x, subtract 2 from both sides of the equation: x = 5 - 2, which gives x = 3.

From the first equation, x = 10 - 2y. Substituting into the second gives 2(10 - 2y) - y = 3, solving gives y = 4, x = 2.

Rearranging gives 4 < x, thus the minimum integer value is 5.

To find x, subtract 3 from both sides: x = 7 - 3, which gives x = 4.

x > 6, so the minimum integer value of x is 7.

Rearranging gives 4 + 1 < 2x - x, thus 5 < x, or x > 5.

Rearranging gives 4 + 3 < x, thus x > 7.

Subtracting 4 from both sides gives x = 6.

We know that x^2 + y^2 = (x + y)^2 - 2xy. Substituting the values, we get (10)^2 - 2(21) = 100 - 42 = 58.

x² + y² = (x + y)² - 2xy = 10² - 2*21 = 100 - 42 = 58.

From x - 2 < 3, we get x < 5. From x + 1 > 0, we get x > -1. Thus, 0 < x < 5.