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In vector notation, **i** and **j** refer to the **unit vectors** along the x-axis and y-axis, respectively. A unit vector has a magnitude of 1 and points in a specific direction. 

- **i** represents the unit vector in the direction of the positive x-axis: \(\mathbf{i} = \langle 1, 0 \rangle\)
- **j** represents the unit vector in the direction of the positive y-axis: \(\mathbf{j} = \langle 0, 1 \rangle\)

For the given vector \( \mathbf{u} = \langle 1, 4 \rangle \), this can be expressed in terms of **i** and **j** as:

\[
\mathbf{u} = 1\mathbf{i} + 4\mathbf{j}
\]

Or more simply:

\[
\mathbf{u} = \mathbf{i} + 4\mathbf{j}
\]

This notation breaks the vector into its horizontal (x-axis) and vertical (y-axis) components using the unit vectors **i** and **j**.

Would you like further details or clarification?

Here are 5 related questions to explore:
1. How do you represent three-dimensional vectors using unit vectors?
2. What is the significance of unit vectors in physics and engineering?
3. How do you calculate the magnitude of a vector given in terms of **i** and **j**?
4. What happens when you add two vectors expressed in terms of **i** and **j**?
5. How can you use **i** and **j** to describe vector direction?

**Tip:** Unit vectors are essential for breaking down vectors into components, making them easier to manipulate in calculations like addition, subtraction, or projections.

Learn how to express vectors in terms of i and j, the unit vectors along the x and y axes, respectively. This solution breaks down the vector u = <1, 4> into its components using i and j, making it easier to visualize and solve vector problems. Ideal for high school students in Grades 9-12.

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