1) Let A = (1,2), B = (-3,4) , then A+B=(-2,6), A-B = (4,-2), -3A=(-3,-6)
2) Let X = (1, 0, π, 4), Y = (2, 4,-2π,-6), then 2X+Y = (4, 4, 0, 2) and
X-(1/2) Y = (0,-2, 2π, 7).
3) Let A = (2, -1), B = (3, 4) , then A+B=(5,3), A-B = (-1,-5), -3A=(-6,3)
4) Let X = (-2, 3, 5), Y = (1, -4, 2), then 2X+Y = (1, 2, 12) and
X-(1/2) Y = (-2, 5, 4)
1) To find the sum of vectors A and B, we need to add their corresponding components:
A + B = (1 + (-3), 2 + 4)
= (-2, 6)
To find the difference between vectors A and B, we need to subtract their corresponding components:
A - B = (1 - (-3), 2 - 4)
= (4, -2)
To find -3A, we need to multiply each component of vector A by -3:
-3A = (-3 * 1, -3 * 2)
= (-3, -6)
Therefore, we have:
A + B = (-2, 6)
A - B = (4, -2)
-3A = (-3, -6)
2) To find 2X + Y, we need to multiply each component of X by 2 and then add them with the corresponding components of Y:
2X + Y = (2 * 1 + 2, 2 * 0 + 4, 2 * π + (-2π), 2 * 4 + (-6))
= (4, 4, 0, 2)
To find X - (1/2)Y, we need to subtract from each component of X, half of the corresponding components of Y:
X - (1/2)Y = (1 - (1/2) * 2, 0 - (1/2) * 4, π - (1/2) * (-2π), 4 - (1/2) * (-6))
= (1 - 1, 0 - 2, π + π, 4 + 3)
= (0, -2, 2π, 7)
Therefore, we have:
2X + Y = (4, 4, 0, 2)
X - (1/2)Y = (0, -2, 2π, 7)