Given x⃗ =2i+4j−3k
, y⃗ =−5i−7j+k
, and z⃗ =8i+5j−9k
,

find :  y⃗ −6x⃗ +2z⃗

To find \( \mathbf{y} - 6\mathbf{x} + 2\mathbf{z} \), we need to perform vector addition and scalar multiplication on the given vectors.

Given vectors:
\[
\mathbf{x} = 2\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}
\]
\[
\mathbf{y} = -5\mathbf{i} - 7\mathbf{j} + \mathbf{k}
\]
\[
\mathbf{z} = 8\mathbf{i} + 5\mathbf{j} - 9\mathbf{k}
\]

First, compute \( 6\mathbf{x} \):
\[
6\mathbf{x} = 6(2\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}) = 12\mathbf{i} + 24\mathbf{j} - 18\mathbf{k}
\]

Next, compute \( 2\mathbf{z} \):
\[
2\mathbf{z} = 2(8\mathbf{i} + 5\mathbf{j} - 9\mathbf{k}) = 16\mathbf{i} + 10\mathbf{j} - 18\mathbf{k}
\]

Now, form the expression \( \mathbf{y} - 6\mathbf{x} + 2\mathbf{z} \):
\[
\mathbf{y} - 6\mathbf{x} + 2\mathbf{z} = (-5\mathbf{i} - 7\mathbf{j} + \mathbf{k}) - (12\mathbf{i} + 24\mathbf{j} - 18\mathbf{k}) + (16\mathbf{i} + 10\mathbf{j} - 18\mathbf{k})
\]

Combine the components for each direction:
- For \( \mathbf{i} \):
\[
-5 - 12 + 16 = -1
\]
- For \( \mathbf{j} \):
\[
-7 - 24 + 10 = -21
\]
- For \( \mathbf{k} \):
\[
1 + 18 - 18 = 1
\]

Therefore, the resulting vector is:
\[
\mathbf{y} - 6\mathbf{x} + 2\mathbf{z} = -\mathbf{i} - 21\mathbf{j} + \mathbf{k}
\]

So, \( \mathbf{y} - 6\mathbf{x} + 2\mathbf{z} = -\mathbf{i} - 21\mathbf{j} + \mathbf{k} \).

Would you like any more details or have any questions about this solution? 

Here are some related questions:

1. How do you perform scalar multiplication on a vector?
2. What is vector addition and how is it done component-wise?
3. How can we represent vectors graphically?
4. What are some real-life applications of vector operations?
5. How does the distributive property apply to vector addition?
6. What is the importance of unit vectors in vector operations?
7. How do you find the magnitude of a vector?
8. What is a dot product and how is it calculated?

**Tip:** When performing vector addition or scalar multiplication, always ensure to handle each component (i, j, k) separately to avoid mistakes.

Learn how to calculate the vector y⃗ −6x⃗ +2z⃗ using vector addition and scalar multiplication. Understand the step-by-step process with detailed explanations and component-wise computations. Suitable for advanced high school students and anyone interested in vector operations.

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