Linear Combinations and Scalar Multiplications

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This worksheet is for 12th-grade math students. Find the required scalar multiples and linear combinations using component equations.

If vector a = (3, -4), find the components of -2a.
Determine whether u = (4, -6) and v = (-10, 15) are parallel.
Determine whether u = (2, 5) and v = (4, 9) are parallel.
Express w = (7, 11) as a linear combination of u = (1, 2) and v = (3, 1). Find c₁ and c₂ where w = c₁u + c₂v.
Express w = (5, 0) as a linear combination of basis vectors i = (1, 0) and j = (0, 1).
Given u = (1, 1) and v = (-1, 1), write w = (2, 6) in the form c₁u + c₂v.

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Linear Combinations and Scalar Multiplications

Answer key

If vector a = (3, -4), find the components of -2a.
(-6, 8)
Determine whether u = (4, -6) and v = (-10, 15) are parallel.
Yes, they are parallel because v = -2.5u.
Determine whether u = (2, 5) and v = (4, 9) are parallel.
No, because 4/2 does not equal 9/5.
Express w = (7, 11) as a linear combination of u = (1, 2) and v = (3, 1). Find c₁ and c₂ where w = c₁u + c₂v.
c₁ = 5 and c₂ = 2/3 (from 7 = c₁ + 3c₂ and 11 = 2c₁ + c₂).
Express w = (5, 0) as a linear combination of basis vectors i = (1, 0) and j = (0, 1).
w = 5i + 0j
Given u = (1, 1) and v = (-1, 1), write w = (2, 6) in the form c₁u + c₂v.
c₁ = 4 and c₂ = 2, so w = 4u + 2v.