What is the length of projection of the vector \(\rm \hat{i}+2 \hat{j}+3 \hat{k}\) on the vector \(\rm2 \hat{i}+3 \hat{j}-2 \hat{k}\) ?
Calculating Vector Projection Length
The question asks for the length of the projection of a vector \(\vec{a} = \hat{i}+2 \hat{j}+3 \hat{k}\) onto another vector \(\vec{b} = 2 \hat{i}+3 \hat{j}-2 \hat{k}\).
To find the length of the projection of vector \(\vec{a}\) onto vector \(\vec{b}\), we use the formula:
Length of projection of \(\vec{a}\) onto \(\vec{b}\) = \(\frac{|\vec{a} \cdot \vec{b}|}{||\vec{b}||}\)
Where:
- \(\vec{a} \cdot \vec{b}\) is the dot product of vectors \(\vec{a}\) and \(\vec{b}\).
- \(||\vec{b}||\) is the magnitude of vector \(\vec{b}\).
Step 1: Calculate the Dot Product (\(\vec{a} \cdot \vec{b}\))
The dot product of two vectors \(\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}\) and \(\vec{b} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}\) is given by \(a_1 b_1 + a_2 b_2 + a_3 b_3\).
Given \(\vec{a} = \hat{i}+2 \hat{j}+3 \hat{k}\) and \(\vec{b} = 2 \hat{i}+3 \hat{j}-2 \hat{k}\), the dot product is:
\(\vec{a} \cdot \vec{b} = (1)(2) + (2)(3) + (3)(-2)\)
\(\vec{a} \cdot \vec{b} = 2 + 6 - 6\)
\(\vec{a} \cdot \vec{b} = 2\)
Step 2: Calculate the Magnitude of \(\vec{b}\) (\(|| \vec{b} ||\))
The magnitude of a vector \(\vec{b} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}\) is given by \(\sqrt{b_1^2 + b_2^2 + b_3^2}\).
Given \(\vec{b} = 2 \hat{i}+3 \hat{j}-2 \hat{k}\), the magnitude is:
\(\|\vec{b}\| = \sqrt{(2)^2 + (3)^2 + (-2)^2}\)
\(\|\vec{b}\| = \sqrt{4 + 9 + 4}\)
\(\|\vec{b}\| = \sqrt{17}\)
Step 3: Calculate the Length of Projection
Now, substitute the calculated values into the projection formula:
Length of projection = \(\frac{|\vec{a} \cdot \vec{b}|}{||\vec{b}||} = \frac{|2|}{\sqrt{17}}\)
Length of projection = \(\frac{2}{\sqrt{17}}\)
Therefore, the length of the projection of the vector \(\hat{i}+2 \hat{j}+3 \hat{k}\) on the vector \(\rm2 \hat{i}+3 \hat{j}-2 \hat{k}\) is \(\frac{2}{\sqrt{17}}\).
Summary of Calculations
| Vector | Components |
|---|---|
| \(\vec{a}\) | \(\hat{i}+2 \hat{j}+3 \hat{k}\) |
| \(\vec{b}\) | \(2 \hat{i}+3 \hat{j}-2 \hat{k}\) |
| Calculation | |
| Dot Product \(\vec{a} \cdot \vec{b}\) | \((1)(2) + (2)(3) + (3)(-2) = 2 + 6 - 6 = 2\) |
| Magnitude \(||\vec{b}||\) | \(\sqrt{2^2 + 3^2 + (-2)^2} = \sqrt{4 + 9 + 4} = \sqrt{17}\) |
| Projection Length | \(\frac{|\vec{a} \cdot \vec{b}|}{||\vec{b}||} = \frac{|2|}{\sqrt{17}} = \frac{2}{\sqrt{17}}\) |
Understanding Vector Projection
The projection of a vector \(\vec{a}\) onto a vector \(\vec{b}\) is the component of \(\vec{a}\) that lies in the direction of \(\vec{b}\). The length of this projection tells us how much of vector \(\vec{a}\) points along the direction of vector \(\vec{b}\).
Revision Table: Key Concepts
| Concept | Definition/Formula |
|---|---|
| Dot Product (\(\vec{a} \cdot \vec{b}\)) | Scalar value: \(\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta\) or \(a_x b_x + a_y b_y + a_z b_z\) |
| Vector Magnitude (\(|| \vec{v} ||\)) | Length of the vector: \(\sqrt{v_x^2 + v_y^2 + v_z^2}\) |
| Projection of \(\vec{a}\) onto \(\vec{b}\) | Vector component of \(\vec{a}\) along \(\vec{b}\): \(\text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{||\vec{b}||^2} \vec{b}\) |
| Length of Projection of \(\vec{a}\) onto \(\vec{b}\) | Scalar value: \(\frac{|\vec{a} \cdot \vec{b}|}{||\vec{b}||}\) |
Additional Information: Applications of Vector Projection
Vector projection is a fundamental concept in linear algebra and physics with various applications, such as:
- Physics: Calculating the work done by a force (Force \(\cdot\) Displacement), resolving forces into components along specific directions.
- Computer Graphics: Determining how a 3D object appears on a 2D screen (orthogonal projection).
- Data Analysis: Dimensionality reduction techniques like Principal Component Analysis (PCA) use concepts related to projections.
- Engineering: Analyzing forces, stresses, and strains in structures.
Understanding how to calculate vector projections and their lengths is crucial for solving problems involving vectors and their interactions in space.
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