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Given two vectors a=i+2j-3k b=2i-3j+4k1. find the magnitude 2. Addition 3.subtraction 4.3a+2b 5.(a.b)6.the angle between the two vectors 7.a×b

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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.

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