Direction: Consider the following for the next two (02) items that follow:
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:
Neither 1 nor 2
Understanding the Problem: Vector Decomposition
The question asks us to analyze a vector \(\vec b\) that is decomposed into two components, \(\vec c\) and \(\vec d\). We are given the original vector \(\vec a\) and \(\vec b\). We know that \(\vec c\) is parallel to \(\vec a\) and \(\vec d\) is perpendicular to \(\vec a\). We are also given the form of vector \(\vec d\) and need to check the correctness of two specific equations involving its components \(x, y,\) and \(z\).
Key Concepts
- Vector Parallelism: If two vectors are parallel, one is a scalar multiple of the other.
- Vector Perpendicularity: If two vectors are perpendicular, their dot product is zero.
- Vector Decomposition: A vector can be expressed as the sum of components parallel and perpendicular to another given vector.
Step-by-Step Solution
We are given the following vectors:
\(\vec a = \hat i + \hat j\)
\(\vec b = 3\hat i + 4\hat k\)
We are told that \(\vec b = \vec c + \vec d\), where \(\vec c\) is parallel to \(\vec a\) and \(\vec d\) is perpendicular to \(\vec a\).
Step 1: Express \(\vec c\) in terms of \(\vec a\)
Since \(\vec c\) is parallel to \(\vec a\), we can write \(\vec c = k \vec a\) for some scalar \(k\). Therefore,
\(\vec c = k(\hat i + \hat j) = k\hat i + k\hat j\)
Step 2: Express \(\vec d\) in terms of \(\vec b\) and \(\vec c\)
From \(\vec b = \vec c + \vec d\), we have \(\vec d = \vec b - \vec c\). Substituting the given \(\vec b\) and our expression for \(\vec c\):
\(\vec d = (3\hat i + 0\hat j + 4\hat k) - (k\hat i + k\hat j + 0\hat k)\)
\(\vec d = (3-k)\hat i - k\hat j + 4\hat k\)
Step 3: Use the perpendicularity condition to find the scalar \(k\)
Since \(\vec d\) is perpendicular to \(\vec a\), their dot product must be zero: \(\vec d \cdot \vec a = 0\).
\(( (3-k)\hat i - k\hat j + 4\hat k ) \cdot ( \hat i + \hat j + 0\hat k ) = 0\)
Taking the dot product:
\((3-k)(1) + (-k)(1) + (4)(0) = 0\)
\(3 - k - k = 0\)
\(3 - 2k = 0\)
\(2k = 3\)
\(k = \frac{3}{2}\)
Step 4: Substitute the value of \(k\) back into the expression for \(\vec d\)
Now that we have the value of \(k\), we can find the specific components of \(\vec d\):
\(\vec d = (3 - \frac{3}{2})\hat i - \frac{3}{2}\hat j + 4\hat k\)
\(\vec d = (\frac{6-3}{2})\hat i - \frac{3}{2}\hat j + 4\hat k\)
\(\vec d = \frac{3}{2}\hat i - \frac{3}{2}\hat j + 4\hat k\)
Step 5: Compare the components of \(\vec d\) with the given form
We are given that \(\vec d = x\hat i + y\hat j + z\hat k\). Comparing this with our calculated \(\vec d\):
\(x = \frac{3}{2}\)
\(y = -\frac{3}{2}\)
\(z = 4\)
Step 6: Check the correctness of the given equations
Now we evaluate the two equations using the values of \(x, y,\) and \(z\).
Equation 1: \(y - x = 4\)
Substitute \(y = -\frac{3}{2}\) and \(x = \frac{3}{2}\):
\(-\frac{3}{2} - \frac{3}{2} = \frac{-3-3}{2} = \frac{-6}{2} = -3\)
We check if \(-3 = 4\). This is false.
Equation 1 is incorrect.
Equation 2: \(2z - 3 = 0\)
Substitute \(z = 4\):
\(2(4) - 3 = 8 - 3 = 5\)
We check if \(5 = 0\). This is false.
Equation 2 is incorrect.
Conclusion
Since neither Equation 1 nor Equation 2 is correct, the correct answer is that neither of the given equations is correct.
| Equation | Calculation | Result | Correct? |
|---|---|---|---|
| \(y - x = 4\) | \(-\frac{3}{2} - \frac{3}{2} = -3\) | \(-3 = 4\) | No |
| \(2z - 3 = 0\) | \(2(4) - 3 = 5\) | \(5 = 0\) | No |
Revision Table: Vector Operations and Decomposition
| Concept | Description | Mathematical Representation |
|---|---|---|
| Vector Addition | Combining two vectors by adding their corresponding components. Geometrically, head-to-tail rule. | \(\vec A + \vec B = (A_x+B_x)\hat i + (A_y+B_y)\hat j + (A_z+B_z)\hat k\) |
| Scalar Multiplication | Multiplying a vector by a scalar changes its magnitude but not direction (unless scalar is negative). | \(k\vec A = kA_x\hat i + kA_y\hat j + kA_z\hat k\) |
| Dot Product | A scalar value representing the projection of one vector onto another. Used to find angle or check perpendicularity. | \(\vec A \cdot \vec B = A_xB_x + A_yB_y + A_zB_z\) |
| Perpendicular Vectors | Two non-zero vectors are perpendicular if their dot product is zero. | \(\vec A \cdot \vec B = 0\) |
| Vector Projection | The component of one vector along the direction of another. | Projection of \(\vec B\) onto \(\vec A\): \({\rm Proj}_{\vec A} \vec B = (\frac{\vec A \cdot \vec B}{||\vec A||^2})\vec A\) |
Additional Information: Vector Decomposition into Parallel and Perpendicular Components
Any vector \(\vec b\) can be uniquely decomposed into two components relative to another non-zero vector \(\vec a\): one component \(\vec c\) parallel to \(\vec a\) and another component \(\vec d\) perpendicular to \(\vec a\). That is, \(\vec b = \vec c + \vec d\).
The component parallel to \(\vec a\) is the vector projection of \(\vec b\) onto \(\vec a\):
\(\vec c = {\rm Proj}_{\vec a} \vec b = (\frac{\vec a \cdot \vec b}{||\vec a||^2})\vec a\)
The component perpendicular to \(\vec a\) can then be found by subtracting the parallel component from the original vector:
\(\vec d = \vec b - \vec c = \vec b - (\frac{\vec a \cdot \vec b}{||\vec a||^2})\vec a\)
In our problem, we used a slightly different approach by setting \(\vec c = k\vec a\) and using the perpendicularity condition \(\vec d \cdot \vec a = 0\) to solve for \(k\). Both methods should yield the same result for \(\vec d\).
Let's verify our result for \(\vec d\) using the projection formula:
\(\vec a = \hat i + \hat j\)
\(\vec b = 3\hat i + 4\hat k\)
\(\vec a \cdot \vec b = (1)(3) + (1)(0) + (0)(4) = 3\)
\(||\vec a||^2 = 1^2 + 1^2 + 0^2 = 1 + 1 = 2\)
Component parallel to \(\vec a\):
\(\vec c = (\frac{3}{2})(\hat i + \hat j) = \frac{3}{2}\hat i + \frac{3}{2}\hat j\)
Component perpendicular to \(\vec a\):
\(\vec d = \vec b - \vec c = (3\hat i + 4\hat k) - (\frac{3}{2}\hat i + \frac{3}{2}\hat j)\)
\(\vec d = (3 - \frac{3}{2})\hat i - \frac{3}{2}\hat j + 4\hat k\)
\(\vec d = \frac{3}{2}\hat i - \frac{3}{2}\hat j + 4\hat k\)
This matches the \(\vec d\) we calculated earlier, confirming our values for \(x, y,\) and \(z\).
Important Questions from Vector Algebra
What is the length of projection of the vector \(\rm \hat{i}+2 \hat{j}+3 \hat{k}\) on the vector \(\rm2 \hat{i}+3 \hat{j}-2 \hat{k}\) ?
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 \(\overrightarrow{A B}\) ?
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?
