91Ó°ÊÓ

Vectors often have different dimensions, and each dimension is referred to as a vector component. Think of a vector as a directional arrow in a coordinate system. The arrow has pieces that extend in the direction of each coordinate axis. Generally, these components are represented as multiples of the unit vectors, like \( \mathbf{i}\) and \( \mathbf{j}\) in two dimensions, which correspond to the x-axis and y-axis respectively.
For the vector \( \mathbf{v} = -3 \mathbf{i} + 7 \mathbf{j} \), it consists of two components:

• A component along the x-axis: \(-3\mathbf{i}\)
• A component along the y-axis: \(7\mathbf{j}\)

These components signify the extent of the vector along each direction. By understanding vector components, you can break down complex vector operations into simpler arithmetic along each axis.

A scalar is just a regular old number, not a vector. When we say "scalar factor," we're referring to multiplying this number by a vector. It's like blowing up or shrinking the vector without changing its direction. The scalar factor acts evenly across all components of the vector, much like scaling the dimensions of a shape in geometry.
For instance, the operation of finding \( 6 \mathbf{v} \) involves multiplying each component of \( \mathbf{v} \) by the scalar 6. This leads to:

• For the x-component: \( 6 \times (-3\mathbf{i}) = -18\mathbf{i} \)
• For the y-component: \( 6 \times 7\mathbf{j} = 42\mathbf{j} \)

The vector's direction remains the same, but it gets longer or shorter based on the scalar value.

The distributive property is a fundamental algebraic principle that also applies in the context of scalar multiplication of vectors. It states that when a scalar multiplies a vector, it is equivalent to multiplying that scalar with each component of the vector separately and then combining the results.
In our exercise, the vector \( \mathbf{v} \) is \( -3 \mathbf{i} + 7 \mathbf{j} \). To apply the scalar factor of 6 using the distributive property, distribute the 6 across each term:

• Multiply the scalar 6 by the x-component \(-3\), yielding \(-18\mathbf{i}\)
• Multiply the scalar 6 by the y-component \(7\), yielding \(42\mathbf{j}\)

Add these results to obtain the scaled vector: \(-18\mathbf{i} + 42\mathbf{j}\). This process simplifies what might initially appear complex, showing the power of algebraic principles like the distributive property.