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.
Important Questions from Vector Algebra
Consider the following in respect of the vectors \(\rm \vec{a}=(0,1,1)\) and \(\rm \vec{b}=(1,0,1) \) :
1. The number of unit vectors perpendicular to both \(\rm \vec{a}\) and \(\rm \vec{b}\) is only one.
2. The angle between the vectors is \(\frac{\pi}{3}\).
Which of the statements given above is/are correct?
Consider the following points :
1. (-1, -3, 1)
2. (-1, 3, 2)
3. (-2, 5, 3)
Which of the above points lie on the line joining A and B ?
What is the magnitude of \(\overrightarrow{A B}\) ?
If \({\rm{\vec d}} = {\rm{x\hat i}} + {\rm{y\hat j}} + {\rm{z\hat k}}\) , then which of the following equations is/are correct?
1. y – x = 4
2. 2z – 3 = 0
Select the correct answer using the code given below:
What is \({\rm{\vec a}} \cdot {\rm{\vec b}} + {\rm{\vec b}} \cdot {\rm{\vec c}} + {\rm{\vec c}} \cdot {\rm{\vec a}}\) equal to?
