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Sagot :
Certainly! Let's solve the problem step by step.
Given vectors:
[tex]\[ v = 4\mathbf{i} + 5\mathbf{j} \][/tex]
[tex]\[ w = -2\mathbf{i} + 3\mathbf{j} \][/tex]
### 1. Find the dot product [tex]\( v \cdot w \)[/tex]:
[tex]\[ v \cdot w = (4)(-2) + (5)(3) = -8 + 15 = 7 \][/tex]
### 2. Find the dot product [tex]\( w \cdot w \)[/tex] (this is the magnitude squared of [tex]\( w \)[/tex]):
[tex]\[ w \cdot w = (-2)^2 + (3)^2 = 4 + 9 = 13 \][/tex]
### 3. Calculate the projection of [tex]\( v \)[/tex] onto [tex]\( w \)[/tex], denoted as [tex]\( \operatorname{proj}_{w} v \)[/tex]:
The formula for projection is:
[tex]\[ \operatorname{proj}_{w} v = \left( \frac{v \cdot w}{w \cdot w} \right) w \][/tex]
First, find the scalar multiplier:
[tex]\[ \frac{v \cdot w}{w \cdot w} = \frac{7}{13} \][/tex]
Then multiply this scalar by the vector [tex]\( w \)[/tex]:
[tex]\[ \operatorname{proj}_{w} v = \frac{7}{13} w = \frac{7}{13} (-2\mathbf{i} + 3\mathbf{j}) = \left( \frac{7}{13} \times -2 \right)\mathbf{i} + \left( \frac{7}{13} \times 3 \right)\mathbf{j} \][/tex]
Calculate the components separately:
[tex]\[ \frac{7}{13} \times -2 = -1.0769230769230769 \][/tex]
[tex]\[ \frac{7}{13} \times 3 = 1.6153846153846154 \][/tex]
So,
[tex]\[ \operatorname{proj}_{w} v = -1.0769230769230769\mathbf{i} + 1.6153846153846154\mathbf{j} \][/tex]
### 4. Decompose [tex]\( v \)[/tex]:
To decompose [tex]\( v \)[/tex] into the sum of its projection onto [tex]\( w \)[/tex] and the perpendicular component, we use:
[tex]\[ v = \operatorname{proj}_{w} v + \text{(perpendicular component)} \][/tex]
The perpendicular component is:
[tex]\[ \text{perpendicular component} = v - \operatorname{proj}_{w} v \][/tex]
Subtract the projection from [tex]\( v \)[/tex]:
[tex]\[ v - \operatorname{proj}_{w} v = (4\mathbf{i} + 5\mathbf{j}) - (-1.0769230769230769\mathbf{i} + 1.6153846153846154\mathbf{j}) \][/tex]
[tex]\[ = (4 + 1.0769230769230769)\mathbf{i} + (5 - 1.6153846153846154)\mathbf{j} \][/tex]
Calculate the components separately:
[tex]\[ 4 + 1.0769230769230769 = 5.076923076923077 \][/tex]
[tex]\[ 5 - 1.6153846153846154 = 3.3846153846153846 \][/tex]
Thus, the perpendicular component is:
[tex]\[ 5.076923076923077\mathbf{i} + 3.3846153846153846\mathbf{j} \][/tex]
### Summary:
- The projection of [tex]\( v \)[/tex] onto [tex]\( w \)[/tex], [tex]\( \operatorname{proj}_{w} v \)[/tex], is:
[tex]\[ -1.0769230769230769\mathbf{i} + 1.6153846153846154\mathbf{j} \][/tex]
- The perpendicular component of [tex]\( v \)[/tex] relative to [tex]\( w \)[/tex] is:
[tex]\[ 5.076923076923077\mathbf{i} + 3.3846153846153846\mathbf{j} \][/tex]
So, we have decomposed [tex]\( v \)[/tex] as:
[tex]\[ v = \operatorname{proj}_{w} v + \text{(perpendicular component)} \][/tex]
[tex]\[ v = (-1.0769230769230769\mathbf{i} + 1.6153846153846154\mathbf{j}) + (5.076923076923077\mathbf{i} + 3.3846153846153846\mathbf{j}) \][/tex]
Given vectors:
[tex]\[ v = 4\mathbf{i} + 5\mathbf{j} \][/tex]
[tex]\[ w = -2\mathbf{i} + 3\mathbf{j} \][/tex]
### 1. Find the dot product [tex]\( v \cdot w \)[/tex]:
[tex]\[ v \cdot w = (4)(-2) + (5)(3) = -8 + 15 = 7 \][/tex]
### 2. Find the dot product [tex]\( w \cdot w \)[/tex] (this is the magnitude squared of [tex]\( w \)[/tex]):
[tex]\[ w \cdot w = (-2)^2 + (3)^2 = 4 + 9 = 13 \][/tex]
### 3. Calculate the projection of [tex]\( v \)[/tex] onto [tex]\( w \)[/tex], denoted as [tex]\( \operatorname{proj}_{w} v \)[/tex]:
The formula for projection is:
[tex]\[ \operatorname{proj}_{w} v = \left( \frac{v \cdot w}{w \cdot w} \right) w \][/tex]
First, find the scalar multiplier:
[tex]\[ \frac{v \cdot w}{w \cdot w} = \frac{7}{13} \][/tex]
Then multiply this scalar by the vector [tex]\( w \)[/tex]:
[tex]\[ \operatorname{proj}_{w} v = \frac{7}{13} w = \frac{7}{13} (-2\mathbf{i} + 3\mathbf{j}) = \left( \frac{7}{13} \times -2 \right)\mathbf{i} + \left( \frac{7}{13} \times 3 \right)\mathbf{j} \][/tex]
Calculate the components separately:
[tex]\[ \frac{7}{13} \times -2 = -1.0769230769230769 \][/tex]
[tex]\[ \frac{7}{13} \times 3 = 1.6153846153846154 \][/tex]
So,
[tex]\[ \operatorname{proj}_{w} v = -1.0769230769230769\mathbf{i} + 1.6153846153846154\mathbf{j} \][/tex]
### 4. Decompose [tex]\( v \)[/tex]:
To decompose [tex]\( v \)[/tex] into the sum of its projection onto [tex]\( w \)[/tex] and the perpendicular component, we use:
[tex]\[ v = \operatorname{proj}_{w} v + \text{(perpendicular component)} \][/tex]
The perpendicular component is:
[tex]\[ \text{perpendicular component} = v - \operatorname{proj}_{w} v \][/tex]
Subtract the projection from [tex]\( v \)[/tex]:
[tex]\[ v - \operatorname{proj}_{w} v = (4\mathbf{i} + 5\mathbf{j}) - (-1.0769230769230769\mathbf{i} + 1.6153846153846154\mathbf{j}) \][/tex]
[tex]\[ = (4 + 1.0769230769230769)\mathbf{i} + (5 - 1.6153846153846154)\mathbf{j} \][/tex]
Calculate the components separately:
[tex]\[ 4 + 1.0769230769230769 = 5.076923076923077 \][/tex]
[tex]\[ 5 - 1.6153846153846154 = 3.3846153846153846 \][/tex]
Thus, the perpendicular component is:
[tex]\[ 5.076923076923077\mathbf{i} + 3.3846153846153846\mathbf{j} \][/tex]
### Summary:
- The projection of [tex]\( v \)[/tex] onto [tex]\( w \)[/tex], [tex]\( \operatorname{proj}_{w} v \)[/tex], is:
[tex]\[ -1.0769230769230769\mathbf{i} + 1.6153846153846154\mathbf{j} \][/tex]
- The perpendicular component of [tex]\( v \)[/tex] relative to [tex]\( w \)[/tex] is:
[tex]\[ 5.076923076923077\mathbf{i} + 3.3846153846153846\mathbf{j} \][/tex]
So, we have decomposed [tex]\( v \)[/tex] as:
[tex]\[ v = \operatorname{proj}_{w} v + \text{(perpendicular component)} \][/tex]
[tex]\[ v = (-1.0769230769230769\mathbf{i} + 1.6153846153846154\mathbf{j}) + (5.076923076923077\mathbf{i} + 3.3846153846153846\mathbf{j}) \][/tex]
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