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

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.

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

A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.

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.

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

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

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

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

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.

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

|-3| = 3, so |x| + x = 3 - 3 = 0.

The absolute value |x| of -5 is 5.

x² + y² = 3² + 4² = 9 + 16 = 25.

2x - 3 = 2(4) - 3 = 8 - 3 = 5.

3x - 2 = 3(4) - 2 = 12 - 2 = 10.

(x - 1)² = (5 - 1)² = 4² = 16.

|z|² = (1)² + (1)² = 1 + 1 = 2.

z^3 = (1 + i)^3 = 1 + 3i + 3i^2 + i^3 = 1 + 3i - 3 - i = -2 + 2i.

The conjugate of a complex number z = a + bi is given by z* = a - bi. Thus, the conjugate of z = 1 + i√3 is 1 - i√3.

z^2 = (1 + i√3)^2 = 1^2 + 2(1)(i√3) + (i√3)^2 = 1 + 2i√3 - 3 = -2 + 2i√3.

The modulus of z = 1 + √3i is |z| = √(1² + (√3)²) = √(1 + 3) = √4 = 2. The square of the modulus is 2² = 4.

z² = (2 + 2i)² = 4 + 8i - 4 = 8i.

The argument of z is given by tan^(-1)(2/2) = tan^(-1)(1) = π/4.