Vector Algebra

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Q1. What is the dot product of vectors A = 2i + 3j and B = 4i - j? (2022)

Solution:

A · B = (2)(4) + (3)(-1) = 8 - 3 = 5.

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Q2. If A = i + 2j + 3k and B = 4i + 5j + 6k, what is A - B? (2023)

Solution:

A - B = (1 - 4)i + (2 - 5)j + (3 - 6)k = -3i - 3j - 3k.

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Q3. If A = 3i + 4j and B = 2i - j, what is A + B? (2021)

Solution:

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

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Q4. What is the projection of vector A = 3i + 4j onto vector B = i + j? (2020)

Solution:

proj_B(A) = (A · B / |B|^2)B = ((3*1 + 4*1) / (1^2 + 1^2))(i + j) = (7/2)(i + j) = 3.5.

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Q5. Find the magnitude of the vector A = 3i - 4j. (2020)

Solution:

|A| = √(3^2 + (-4)^2) = √(9 + 16) = √25 = 5.

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Q6. If A = 2i + 3j and B = 3i + 4j, what is the angle between A and B? (2021)

Solution:

cos(θ) = (A · B) / (|A||B|) = (2*3 + 3*4) / (√(2^2 + 3^2) * √(3^2 + 4^2)) = 0.7071, θ = 45 degrees.

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Q7. What is the cross product of vectors A = i + j and B = j + k? (2022)

Solution:

A × B = |i j k| |1 1 0| |0 1 1| = i(1*1 - 0*1) - j(1*1 - 0*0) + k(1*0 - 1*0) = i - j.

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Q8. If A = 5i + 12j, what is the unit vector in the direction of A? (2021)

Solution:

Unit vector = A / |A| = (5i + 12j) / √(5^2 + 12^2) = (5/13)i + (12/13)j.

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Q9. If A = 4i + 2j and B = -i + 3j, what is the scalar triple product A · (B × A)? (2023)

Solution:

B × A = |i j k| |-1 3 0| |4 2 0| = 0, hence A · (B × A) = 0.