The vectors given are:

$\vec{A} = (1.21) \hat{i} + (4.42) \hat{j} + (8.28) \hat{k}$ $\vec{B} = (-10.04) \hat{i} + (1.29) \hat{j} + (3.81) \hat{k}$ $\vec{C} = (11.59) \hat{i} + (-2.58) \hat{j} + (3.31) \hat{k}$

The vector sum $\vec{V} = \vec{A} + \vec{B} + \vec{C}$ will involve adding the components of $\vec{A}$, $\vec{B}$, and $\vec{C}$ for the $\hat{i}$, $\hat{j}$, and $\hat{k}$ directions separately.

To find the x-component (which corresponds to the $\hat{i}$ direction) of the vector sum $\vec{V}$, we simply add the x-components of $\vec{A}$, $\vec{B}$, and $\vec{C}$.

The x-components are:

• $\vec{A}_x = 1.21$
• $\vec{B}_x = -10.04$
• $\vec{C}_x = 11.59$

Now, summing them:

$V_x = 1.21 + (-10.04) + 11.59$

Let's calculate this.The x-component of the vector sum $\vec{V} = \vec{A} + \vec{B} + \vec{C}$ is approximately $2.76$.

Would you like further details or have any questions?

Here are five related questions:

1. How do you calculate the y-component of the vector sum $\vec{V}$?
2. What is the full vector form of $\vec{V}$?
3. How do you compute the magnitude of a vector from its components?
4. Can you find the dot product of two vectors like $\vec{A} \cdot \vec{B}$?
5. How do you calculate the cross product of two vectors?

Tip: When summing vectors, always handle the components in each direction ($\hat{i}$, $\hat{j}$, $\hat{k}$) independently.