91Ó°ÊÓ

In vector mathematics, basis vectors are a set of vectors that, when combined, can represent any vector in a particular space. These vectors are usually denoted as \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \), which correspond to the unit vectors along the x, y, and z axes respectively. Any vector in three-dimensional space can be expressed in terms of these basis vectors.
Think of these basis vectors as building blocks. Just like you can combine bricks to build a wall, you can use combinations of \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) to construct a vector representing a point in space.
For example, the vector \( \mathbf{a} = 1\mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \) tells us that it is 1 unit along the x-axis, 2 units along the y-axis, and -3 units along the z-axis.

Vector addition involves combining vectors to see how they translate a point in space. It's a straightforward process of adding corresponding components of two vectors. The goal is to find a resultant vector that represents the combined effect of the original vectors.
Suppose you have two vectors: \( \mathbf{a} = 1\mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \) and \( \mathbf{b} = 4\mathbf{i} + 0\mathbf{j} + 7\mathbf{k} \). To add these vectors, add each of their components:

• i-components: \( 1+4 = 5 \)
• j-components: \( 2+0 = 2 \)
• k-components: \( -3+7 = 4 \)

This results in a new vector \( \mathbf{c} = 5\mathbf{i} + 2\mathbf{j} + 4\mathbf{k} \), indicating a movement of 5 units along x, 2 units along y, and 4 units along z.

Vector subtraction is similar to vector addition, but instead involves finding the difference between two vectors. In other words, subtract the corresponding components of one vector from another. This process produces another vector that indicates the relative position of one vector to another.
Using the same vectors \( \mathbf{a} = 1\mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \) and \( \mathbf{b} = 4\mathbf{i} + 0\mathbf{j} + 7\mathbf{k} \), vector subtraction \( \mathbf{a} - \mathbf{b} \) is calculated as:

• i-components: \( 1-4 = -3 \)
• j-components: \( 2-0 = 2 \)
• k-components: \( -3-7 = -10 \)

This results in the vector \( \mathbf{d} = -3\mathbf{i} + 2\mathbf{j} - 10\mathbf{k} \), which shifts you back 3 units along x, up 2 along y, and down 10 along z.

Scalar multiplication involves multiplying a vector by a real number called a scalar. This operation scales the vector by changing its magnitude while preserving its direction, unless the scalar is negative, in which case the direction is reversed.
If we take the vector \( \mathbf{a} = 1\mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \) and scale it by 2, the result is:
\[ 2 \cdot (1\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}) = 2\mathbf{i} + 4\mathbf{j} - 6\mathbf{k} \]
Similarly, for vector \( \mathbf{b} = 4\mathbf{i} + 0\mathbf{j} + 7\mathbf{k} \) scaled by 3:
\[ 3 \cdot (4\mathbf{i} + 0\mathbf{j} + 7\mathbf{k}) = 12\mathbf{i} + 0\mathbf{j} + 21\mathbf{k} \]
Scaling a vector helps in calculations where a vector needs to be extended or compressed, such as in vector addition or subtraction after the scaling process.