Given vectors [tex]\( \mathbf{v} = 4\mathbf{i} + 5\mathbf{j} \)[/tex] and [tex]\( \mathbf{w} = -2\mathbf{i} + 3\mathbf{j} \)[/tex], we need to find the projection of [tex]\( \mathbf{v} \)[/tex] onto [tex]\( \mathbf{w} \)[/tex] and then decompose [tex]\( \mathbf{v} \)[/tex] into its component along [tex]\( \mathbf{w} \)[/tex].
Let's start with the projection calculation:
1. Dot Product:
[tex]\[
\mathbf{v} \cdot \mathbf{w} = (4)(-2) + (5)(3) = -8 + 15 = 7
\][/tex]
2. Magnitude Squared of [tex]\( \mathbf{w} \)[/tex] (also the dot product of [tex]\( \mathbf{w} \)[/tex] with itself):
[tex]\[
\mathbf{w} \cdot \mathbf{w} = (-2)^2 + (3)^2 = 4 + 9 = 13
\][/tex]
3. Projection Formula:
[tex]\[
\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \left( \frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} \right) \mathbf{w} = \left( \frac{7}{13} \right) \mathbf{w} = \left( \frac{7}{13} \right) (-2\mathbf{i} + 3\mathbf{j})
\][/tex]
So,
[tex]\[
\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \left( \frac{7}{13} \right)(-2\mathbf{i} + 3\mathbf{j}) = \left( -\frac{14}{13} \right)\mathbf{i} + \left( \frac{21}{13} \right) \mathbf{j}
\][/tex]
Thus,
[tex]\[
\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \left( -\frac{14}{13}\mathbf{i} + \frac{21}{13}\mathbf{j} \right)
\][/tex]
Simplifying, we round to approximate decimal values:
[tex]\[
\operatorname{proj}_{\mathbf{w}} \mathbf{v} \approx -1.07692308\mathbf{i} + 1.61538462\mathbf{j}
\][/tex]
Let's assign this to [tex]\( v_1\)[/tex]:
[tex]\[
v_1 = \operatorname{proj}_{\mathbf{w}} \mathbf{v} = \left( -\frac{14}{13}\mathbf{i} + \frac{21}{13}\mathbf{j} \right)
\][/tex]
So, in summary:
[tex]\[
\operatorname{proj}_{\mathbf{w}} \mathbf{v} = -\frac{14}{13} \mathbf{i} + \frac{21}{13} \mathbf{j}
\][/tex]
And,
[tex]\[
v_1 = -\frac{14}{13} \mathbf{i} + \frac{21}{13} \mathbf{j}
\][/tex]
Or approximately:
[tex]\[
v_1 \approx -1.07692308 \mathbf{i} + 1.61538462 \mathbf{j}
\][/tex]
