When two lines intersect, they form two pairs of vertically opposite angles. The sum of the angles around a point is 360 degrees, but the angles formed by the intersecting lines sum up to 180 degrees.

When two lines intersect, they form two pairs of vertically opposite angles. The sum of the angles around a point is 360 degrees, and since there are two pairs of vertically opposite angles, the sum of the angles formed at the intersection is 180 degrees.

Alternate interior angles are equal when two parallel lines are cut by a transversal. Therefore, the other angle is also 45 degrees.

Alternate interior angles are equal when two parallel lines are cut by a transversal. Therefore, the other alternate interior angle is also 75 degrees.

When two parallel lines are cut by a transversal, the alternate interior angles are equal.

A × B = |i  j  k|\n|2  3  1|\n|1 -2  4| = (3*4 - 1*(-2))i - (2*4 - 1*1)j + (2*(-2) - 3*1)k = 10i - 5j - 7k.

R = A + B = (2 - 1)i + (3 + 4)j = 1i + 7j.

cos(θ) = (A · B) / (|A| |B|) = (2*3 + 3*4) / (√(2^2 + 3^2) * √(3^2 + 4^2)) = 0.6, θ = 53.13 degrees.

cos(θ) = (A · B) / (|A||B|) = (8 + 15) / (√(13) * √(41)). θ = 60 degrees.

cos(θ) = (A · B) / (|A||B|) = (2*4 + 3*5) / (√(2^2 + 3^2) * √(4^2 + 5^2)) = 0.5, θ = 60 degrees.

Vectors A and B are parallel because B = 2A.

cos(θ) = (A · B) / (|A| |B|). A · B = 8 + 0 + 3(0) = 8; |A| = √(2^2 + 3^2) = √13; |B| = √(4^2 + 1^2) = √17. θ = cos^(-1)(8/(√13 * √17)).

A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.

A × B = |i  j  k| |3  4  0| |4  3  0| = (0 - 0)i - (0 - 0)j + (9 - 12)k = -3k.

cos(θ) = (A · B) / (|A||B|) = (4*-3 + 3*4) / (5*5) = 0, θ = 90 degrees.

A + B = (4 - 1)i + (3 + 2)j = 3i + 5j.

R = A + B = (4 - 1)i + (3 + 2)j = 3i + 5j.

cos(θ) = (A · B) / (|A||B|) = (12 + 12) / (5 * 5) = 24/25, θ = cos^(-1)(24/25) ≈ 60 degrees.

A · B = 16 - 9 = 7. |A| = 5, |B| = 5. cos(θ) = 7/(5*5) = 0.28, θ = 180 degrees.

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.