Determining the Value of P for Perpendicular Vectors

This question asks us to find the value of 'p' given two vectors, $\vec a$ and $\vec b$, which are stated to be perpendicular to each other. Understanding the condition for perpendicular vectors is key to solving this problem.

Condition for Perpendicular Vectors

Two non-zero vectors are perpendicular (or orthogonal) if and only if their dot product (also known as the scalar product) is equal to zero. Mathematically, if $\vec a \perp \vec b$, then $\vec a \cdot \vec b = 0$.

Vector Dot Product Formula

For two vectors given in component form, $\vec a = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$ and $\vec b = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k}$, their dot product is calculated as:

$\vec a \cdot \vec b = a_1b_1 + a_2b_2 + a_3b_3$

Applying the Formula to the Given Vectors

We are given the vectors:

$\vec a = 3\mathbf{i} + 4\mathbf{j} - 7\mathbf{k}$

$\vec b = p\mathbf{i} - 6\mathbf{j} + 3\mathbf{k}$

Here, the components are:

• $a_1 = 3$, $a_2 = 4$, $a_3 = -7$
• $b_1 = p$, $b_2 = -6$, $b_3 = 3$

Since the vectors are perpendicular, their dot product must be zero:

$\vec a \cdot \vec b = (3)(p) + (4)(-6) + (-7)(3) = 0$

Step-by-Step Calculation

Let's compute the dot product and set it to zero:

1. Multiply the corresponding components: $3 \times p = 3p$
2. Multiply the corresponding components: $4 \times (-6) = -24$
3. Multiply the corresponding components: $(-7) \times 3 = -21$
4. Sum these products and set the sum to zero: $3p + (-24) + (-21) = 0$
5. Simplify the equation: $3p - 24 - 21 = 0$
6. Combine the constant terms: $3p - 45 = 0$
7. Isolate the term with 'p': $3p = 45$
8. Solve for 'p' by dividing both sides by 3: $p = \frac{45}{3}$
9. Calculate the final value: $p = 15$

Therefore, the value of p that makes the two vectors perpendicular is 15.