Consider two vectors A = 4i - 4j and B = -i - 8j .

(a) Calculate A + B.
(b) Calculate A - B.
(c) Calculate absolute value(A + B).
(d) Calculate absolute value(A - B).

(e) What is the direction of A + B? (from the +x axis)
(f) What is the direction of A - B? (from the +x axis)

(a) Well, to calculate A + B, we simply add the corresponding components. So we have (4-1)i + (-4-8)j = 3i - 12j.

(b) To calculate A - B, we subtract the corresponding components. So we have (4-(-1))i + (-4-(-8))j = 5i + 4j.

(c) To calculate the absolute value of a vector, we use the Pythagorean theorem. So the absolute value of A + B is sqrt((3^2) + (-12^2)) = sqrt(9 + 144) = sqrt(153).

(d) The absolute value of A - B is sqrt((5^2) + (4^2)) = sqrt(25 + 16) = sqrt(41).

(e) The direction of A + B is determined by arctan(-12/3) = -75.96 degrees or 284.04 degrees from the +x axis.

(f) The direction of A - B is determined by arctan(4/5) = 38.66 degrees or 218.66 degrees from the +x axis.

Hope that gives you a good laugh!

To calculate A + B, we add the corresponding components of A and B:

A + B = (4i - 4j) + (-i - 8j)
= (4 - 1)i + (-4 - 8)j
= 3i - 12j.

So, A + B = 3i - 12j.

To calculate A - B, we subtract the corresponding components of B from A:

A - B = (4i - 4j) - (-i - 8j)
= (4 - (-1))i + (-4 - (-8))j
= 5i + 4j.

So, A - B = 5i + 4j.

To calculate the absolute value of A + B, we take the magnitude of the vector A + B:

|A + B| = √((3)^2 + (-12)^2)
= √(9 + 144)
= √153.

So, |A + B| = √153.

To calculate the absolute value of A - B, we take the magnitude of the vector A - B:

|A - B| = √((5)^2 + (4)^2)
= √(25 + 16)
= √41.

So, |A - B| = √41.

To find the direction of A + B from the +x axis, we use the arctan function:

Direction of A + B = arctan(-12/3)
= arctan(-4).

So, the direction of A + B from the +x axis is approximately -75.96 degrees.

To find the direction of A - B from the +x axis, we use the arctan function:

Direction of A - B = arctan(4/5).

So, the direction of A - B from the +x axis is approximately 38.66 degrees.

To calculate the sum, subtraction, and absolute value of the given vectors A and B, we can use the following steps:

Given vectors:
A = 4i - 4j
B = -i - 8j

(a) To calculate A + B, we add the corresponding components of the two vectors:
A + B = (4i - 4j) + (-i - 8j)
= 4i - i - 4j - 8j
= (4 - 1)i + (-4 - 8)j
= 3i - 12j

(b) To calculate A - B, we subtract the corresponding components of the two vectors:
A - B = (4i - 4j) - (-i - 8j)
= 4i + i - (-4j) - (-8j)
= (4 + 1)i + (-4 + 8)j
= 5i + 4j

(c) To calculate the absolute value (magnitude) of A + B, we can use the Pythagorean theorem:
|A + B| = √[(3i)^2 + (-12j)^2]
= √[9i^2 + 144j^2]
= √(9 + 144)
= √153

(d) To calculate the absolute value (magnitude) of A - B:
|A - B| = √[(5i)^2 + (4j)^2]
= √[25i^2 + 16j^2]
= √(25 + 16)
= √41

(e) To find the direction of A + B (from the +x-axis):
We can calculate the angle θ using the arctangent function:
θ = tan^(-1)(A_y / A_x)
= tan^(-1)(-12 / 3)
= tan^(-1)(-4)
≈ -75.96°

Therefore, the direction of A + B from the +x-axis is approximately -75.96° (counterclockwise).

(f) To find the direction of A - B (from the +x-axis):
Similarly, we calculate the angle θ:
θ = tan^(-1)(A_y / A_x)
= tan^(-1)(4 / 5)
≈ 38.66°

Therefore, the direction of A - B from the +x-axis is approximately 38.66° (counterclockwise).