. Compute the following quantities, if they make sense. If they don’t make any sense, just write “nonsense”. a) (2, −3, 1) + (−7, 5, 2) b) 3(−1, 0, 4) − 2(1, 3, 0) c) 2(1, 4, −2, 5) + (2, 3, 0, 1, 0) d) 4(1, 2) + 3(0, −1) + (5, 6)

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:

\[
(4, 8) + (0, −3) + (5, 6) = (4 + 0 + 5, 8 + (−3) + 6) = (9, 11)
\]

---

**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.

Learn how to compute vector operations including addition and scalar multiplication. Solve examples like (2, −3, 1) + (−7, 5, 2) and 3(−1, 0, 4) − 2(1, 3, 0). Understand why adding vectors of different dimensions is considered 'nonsense'. Suitable for students and enthusiasts interested in linear algebra and vector mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation