Let's go through each of the operations step by step using the given vectors:
Given vectors:
a = i + 2j - 3k
b = 2i - 3j + 4k
Magnitude of vector a:
The magnitude of vector a is given by the formula: |a| = √(a_i² + a_j² + a_k²)
Substitute the values: |a| = √(1² + 2² + (-3)²) = √(1 + 4 + 9) = √14
Magnitude of vector b:
The magnitude of vector b is given by the same formula: |b| = √(b_i² + b_j² + b_k²)
Substitute the values: |b| = √(2² + (-3)² + 4²) = √(4 + 9 + 16) = √29
Vector addition (a + b):
Adding corresponding components of vectors a and b:
a + b = (1 + 2)i + (2 - 3)j + (-3 + 4)k
a + b = 3i - j + k
Vector subtraction (a - b):
Subtracting corresponding components of vectors a and b:
a - b = (1 - 2)i + (2 + 3)j + (-3 - 4)k
a - b = -i + 5j - 7k
Vector scalar multiplication (3a + 2b):
Multiplying each component of vector a by 3 and each component of vector b by 2, then adding them:
3a + 2b = 3(i + 2j - 3k) + 2(2i - 3j + 4k)
3a + 2b = (3i + 6j - 9k) + (4i - 6j + 8k)
3a + 2b = 7i + 0j - k
Dot product (a ⋅ b):
The dot product of vectors a and b is given by the formula: a ⋅ b = a_i * b_i + a_j * b_j + a_k * b_k
Substitute the values: a ⋅ b = (1 * 2) + (2 * -3) + (-3 * 4) = 2 - 6 - 12 = -16
Cross product (a × b):
The cross product of vectors a and b is given by the formula:
a × b = (a_j * b_k - a_k * b_j)i - (a_i * b_k - a_k * b_i)j + (a_i * b_j - a_j * b_i)k
Substitute the values: a × b = (2 * 4 - (-3) * (-3))i - (1 * 4 - (-3) * 2)j + (1 * (-3) - 2 * 2)k
a × b = (8 - 9)i - (4 + 6)j + (-3 - 4)k
a × b = -i - 10j - 7k
Angle between the two vectors:
The angle θ between two vectors a and b can be found using the formula: cos(θ) = (a ⋅ b) / (|a| * |b|)
Substitute the values: cos(θ) = (-16) / (√14 * √29)
θ = acos(-16 / (√14 * √29))
The values of √14 and √29 are approximated here. To find the numerical value, you can use a calculator.