Given the vectors [tex]\( \mathbf{v} = 3\mathbf{i} + 5\mathbf{j} \)[/tex] and [tex]\( \mathbf{w} = -2\mathbf{i} + 3\mathbf{j} \)[/tex], we are asked to find [tex]\( v_1 \)[/tex].
Let's break it down step-by-step:
1. Understanding the Vector Notation:
- The vector [tex]\( \mathbf{v} \)[/tex] is given in component form and can be written as [tex]\( \mathbf{v} = \begin{pmatrix} 3 \\ 5 \end{pmatrix} \)[/tex]. This means that the vector [tex]\( \mathbf{v} \)[/tex] has two parts:
- The [tex]\( i \)[/tex]-component (or [tex]\( x \)[/tex]-component) is 3.
- The [tex]\( j \)[/tex]-component (or [tex]\( y \)[/tex]-component) is 5.
2. Identifying [tex]\( v_1 \)[/tex]:
- [tex]\( v_1 \)[/tex] generally represents the first component (or the [tex]\( x \)[/tex]-component) of the vector [tex]\( \mathbf{v} \)[/tex].
- From the vector [tex]\( \mathbf{v} = 3\mathbf{i} + 5\mathbf{j} \)[/tex], the [tex]\( x \)[/tex]-component (or the first component [tex]\( v_1 \)[/tex]) is 3.
Therefore, based on our understanding of vector components, we find that [tex]\( v_1 = 3 \)[/tex].
Hence, the value of [tex]\( v_1 \)[/tex] is [tex]\( \boxed{3} \)[/tex].
