Let's compute each of the given quantities step by step.

a) $(2, −3, 1) + (−7, 5, 2)$

This is a vector addition problem, where both vectors have the same dimensions (3D vectors).

$(2, −3, 1) + (−7, 5, 2) = (2 + (−7), −3 + 5, 1 + 2) = (−5, 2, 3)$

b) $3(−1, 0, 4) − 2(1, 3, 0)$

This involves scalar multiplication and then vector subtraction. Let's do the scalar multiplications first:

$3(−1, 0, 4) = (3 \times −1, 3 \times 0, 3 \times 4) = (−3, 0, 12)$

$2(1, 3, 0) = (2 \times 1, 2 \times 3, 2 \times 0) = (2, 6, 0)$

Now, subtract the second vector from the first:

$(−3, 0, 12) − (2, 6, 0) = (−3 − 2, 0 − 6, 12 − 0) = (−5, −6, 12)$

c) $2(1, 4, −2, 5) + (2, 3, 0, 1, 0)$

Here, the first vector has 4 dimensions, and the second one has 5 dimensions. You can only add vectors of the same dimension.

Since these vectors have different dimensions, this operation is nonsense.

d) $4(1, 2) + 3(0, −1) + (5, 6)$

This involves scalar multiplication followed by vector addition. Let's compute each step:

$4(1, 2) = (4 \times 1, 4 \times 2) = (4, 8)$

$3(0, −1) = (3 \times 0, 3 \times −1) = (0, −3)$

Now, add all the vectors together:

$(4, 8) + (0, −3) + (5, 6) = (4 + 0 + 5, 8 + (−3) + 6) = (9, 11)$

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Summary of Results:

• a) $(-5, 2, 3)$
• b) $(-5, -6, 12)$
• c) nonsense
• d) $(9, 11)$

Would you like any further details on these calculations or have any other questions?

Relative Questions:

1. What are the general rules for vector addition and scalar multiplication?
2. Can you give an example of adding two 4-dimensional vectors?
3. How would the result change if the scalar in part b was negative?
4. What happens if you add vectors of different dimensions? Why is it considered nonsense?
5. Can you explain how scalar multiplication affects the direction and magnitude of a vector?

Tip:

Always check the dimensions of vectors before performing operations like addition or subtraction to ensure they are compatible.