If \(\overrightarrow a = 2\widehat i + 2\widehat j + 3\widehat k,\overrightarrow b = - \widehat i + 2\widehat j + \hat k \rm \ and\overrightarrow c = 3\widehat i + \widehat j\) are such that \(\overrightarrow a + γ \overrightarrow b \) is perpendicular to \(\overrightarrow c \), then determine the value of γ?
8
Understanding Vector Perpendicularity and Scalar Multiplication
This problem involves understanding fundamental vector operations, including scalar multiplication, vector addition, and the dot product. The core concept to solve this problem is that if two vectors are perpendicular, their dot product must be zero. We need to determine the value of a scalar &(\gamma&) that satisfies this condition for the given vectors.
Given Vectors and the Perpendicularity Condition
We are provided with three vectors in three-dimensional space:
- Vector &(\overrightarrow a&) = \(2\widehat i + 2\widehat j + 3\widehat k\)
- Vector &(\overrightarrow b&) = \( - \widehat i + 2\widehat j + \hat k\)
- Vector &(\overrightarrow c&) = \(3\widehat i + \widehat j\)
The problem states that the vector sum &(\overrightarrow a + γ \overrightarrow b&) is perpendicular to vector &(\overrightarrow c&). Mathematically, the condition for two vectors &(\overrightarrow P&) and &(\overrightarrow Q&) to be perpendicular is that their dot product is zero, i.e., &(\overrightarrow P \cdot \overrightarrow Q = 0&). Therefore, we can write the given condition as:
\((\overrightarrow a + γ \overrightarrow b) \cdot \overrightarrow c = 0\)
Calculating the Sum Vector &(\overrightarrow a + γ \overrightarrow b&)
First, we need to determine the resultant vector &(\overrightarrow a + γ \overrightarrow b&). This involves two steps: scalar multiplication of vector &(\overrightarrow b&) by &(\gamma&), followed by vector addition with &(\overrightarrow a&).
Step 1: Scalar Multiplication of &(\overrightarrow b&) by &(\gamma&)
Multiply each component of &(\overrightarrow b&) by the scalar &(\gamma&):
\(γ \overrightarrow b = γ (- \widehat i + 2\widehat j + \hat k) = - γ \widehat i + 2γ \widehat j + γ \hat k\)
Step 2: Vector Addition of &(\overrightarrow a&) and &(γ \overrightarrow b&))
Now, add the components of &(\overrightarrow a&) and &(γ \overrightarrow b&):
\(\overrightarrow a + γ \overrightarrow b = (2\widehat i + 2\widehat j + 3\widehat k) + (- γ \widehat i + 2γ \widehat j + γ \hat k)\)
Combine the respective &(\widehat i&), &(\widehat j&), and &(\widehat k&) components:
\(\overrightarrow a + γ \overrightarrow b = (2 - γ)\widehat i + (2 + 2γ)\widehat j + (3 + γ)\widehat k\)
Applying the Dot Product for Perpendicular Vectors
Now that we have the expression for &(\overrightarrow a + γ \overrightarrow b&), we will use the perpendicularity condition: the dot product of &(\overrightarrow a + γ \overrightarrow b&) and &(\overrightarrow c&) must be zero. The dot product of two vectors &(\overrightarrow P = P_x\widehat i + P_y\widehat j + P_z\widehat k&) and &(\overrightarrow Q = Q_x\widehat i + Q_y\widehat j + Q_z\widehat k&) is calculated as &(\overrightarrow P \cdot \overrightarrow Q = P_x Q_x + P_y Q_y + P_z Q_z&).
We have:
- &(\overrightarrow a + γ \overrightarrow b&) = \((2 - γ)\widehat i + (2 + 2γ)\widehat j + (3 + γ)\widehat k\)
- &(\overrightarrow c&) = \(3\widehat i + 1\widehat j + 0\widehat k\) (Note that the &(\widehat k&)-component of &(\overrightarrow c&) is zero)
Calculate their dot product:
\((\overrightarrow a + γ \overrightarrow b) \cdot \overrightarrow c = ((2 - γ)\widehat i + (2 + 2γ)\widehat j + (3 + γ)\widehat k) \cdot (3\widehat i + 1\widehat j + 0\widehat k)\)
\(= (2 - γ)(3) + (2 + 2γ)(1) + (3 + γ)(0)\)
Simplify the expression:
\(= 6 - 3γ + 2 + 2γ + 0\)
\(= 8 - γ\)
Determining the Value of γ
As per the perpendicularity condition, the dot product must be equal to zero:
\(8 - γ = 0\)
To find the value of &(\gamma&), we solve this simple linear equation:
\(γ = 8\)
Therefore, the value of &(\gamma&) that makes &(\overrightarrow a + γ \overrightarrow b&) perpendicular to &(\overrightarrow c&) is 8.
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 \(\overrightarrow{A B}\) ?
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:
