Step 1: Understanding the Concept:
Vectors are quantities that possess both magnitude and direction. In 3D space, they are represented by components along the \( X, Y, \) and \( Z \) axes (\( \hat{i}, \hat{j}, \hat{k} \)).
The dot product (scalar product) of two vectors \( \vec{A} \) and \( \vec{B} \) is defined as \( \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta \), where \( \theta \) is the angle between the vectors.
When two vectors are perpendicular (orthogonal), the angle between them is \( 90^\circ \). Since \( \cos 90^\circ = 0 \), the dot product of two perpendicular vectors must always be zero.
This property allows us to solve for unknown components in vector equations by setting the sum of the products of corresponding components to zero.
Step 2: Key Formula or Approach:
For vectors \( \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \), the dot product is calculated as:
\[ \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 \]
The condition for perpendicularity is \( \vec{a} \cdot \vec{b} = 0 \).
Step 3: Detailed Explanation:
The first vector is given as \( \vec{a} = 2\hat{i} + 1\hat{j} + 3\hat{k} \).
The second vector is given as \( \vec{b} = p\hat{i} + 2\hat{j} + 2\hat{k} \).
Applying the perpendicularity condition, we set their dot product to zero:
\[ (2\hat{i} + 1\hat{j} + 3\hat{k}) \cdot (p\hat{i} + 2\hat{j} + 2\hat{k}) = 0 \]
Perform the multiplication for each component separately:
- For the \( \hat{i} \) components: \( 2 \times p = 2p \)
- For the \( \hat{j} \) components: \( 1 \times 2 = 2 \)
- For the \( \hat{k} \) components: \( 3 \times 2 = 6 \)
Summing these products gives the total dot product:
\[ 2p + 2 + 6 = 0 \]
Simplify the linear equation by combining the constant numerical terms:
\[ 2p + 8 = 0 \]
To solve for the variable \( p \), move the constant to the other side of the equation:
\[ 2p = -8 \]
Divide both sides by 2:
\[ p = \frac{-8}{2} \]
\[ p = -4 \]
Step 4: Final Answer:
The dot product calculation results in a simple linear equation.
Solving \( 2p + 8 = 0 \) gives \( p = -4 \).
Thus, the correct value of the parameter \( p \) is \( -4 \).