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.