To solve $v + u + 3w$ with the given vectors, we can follow these steps:

Given Vectors

• $u = \langle 7, -4 \rangle$
• $v = \langle -2, 8 \rangle$
• $w = \langle -2, -5 \rangle$

Step 1: Calculate $3w$

To find $3w$, multiply each component of $w$ by 3: $3w = 3 \cdot \langle -2, -5 \rangle = \langle 3 \cdot -2, 3 \cdot -5 \rangle = \langle -6, -15 \rangle$

Step 2: Calculate $v + u$

Add corresponding components of $v$ and $u$: $v + u = \langle -2, 8 \rangle + \langle 7, -4 \rangle = \langle -2 + 7, 8 - 4 \rangle = \langle 5, 4 \rangle$

Step 3: Calculate $v + u + 3w$

Now add the result of $v + u$ with $3w$: $v + u + 3w = \langle 5, 4 \rangle + \langle -6, -15 \rangle = \langle 5 - 6, 4 - 15 \rangle = \langle -1, -11 \rangle$

Final Answer

$v + u + 3w = \langle -1, -11 \rangle$

Would you like further details or have any questions?

Here are some related questions you might find interesting:

1. How do we generalize the formula for scaling and adding multiple vectors?
2. What would happen if we changed the scalar multiplier for $w$ in this problem?
3. How do vector operations relate to real-world physics applications?
4. What is the geometric interpretation of adding and scaling vectors in two-dimensional space?
5. How would this solution change if the vectors were three-dimensional?

Tip: Remember that multiplying a vector by a scalar scales each of its components by that scalar.