The given expression in the image is:

$\mathbf{a} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$

We are asked to find $a_0$.

In vector notation, a vector $\mathbf{a}$ can be written as:

$\mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k}$

Comparing this with the given vector $\mathbf{a} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$, we can observe that:

• The component along $\mathbf{i}$ (denoted $a_1$) is 1.
• The component along $\mathbf{j}$ (denoted $a_2$) is 2.
• The component along $\mathbf{k}$ (denoted $a_3$) is 3.

Now, the term $a_0$ often refers to the magnitude of the vector $\mathbf{a}$, which can be calculated using the formula for the magnitude of a 3-dimensional vector:

$|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$

Substituting the values $a_1 = 1$, $a_2 = 2$, and $a_3 = 3$, we get:

$|\mathbf{a}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14}$

Thus, $a_0 = |\mathbf{a}| = \sqrt{14}$.

Final Answer:

$a_0 = \sqrt{14}$

Let me know if you'd like any further details or explanations!

Related Questions:

1. What is the geometric interpretation of the magnitude of a vector?
2. How do we generalize the formula for the magnitude of a vector in n-dimensions?
3. Can you find the unit vector corresponding to $\mathbf{a} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$?
4. How does the direction of a vector relate to its components?
5. What happens if one or more of the components of the vector $\mathbf{a}$ were negative?

Tip:

Remember that when dealing with vectors, the magnitude (or norm) represents the "length" of the vector in Euclidean space!