Key Concepts and Formulas

1. Cross Product Property: If $\vec{A} \times \vec{B} = \vec{C} \times \vec{B}$ and $\vec{B} \neq \vec{0}$, then $\vec{A} - \vec{C}$ is parallel to $\vec{B}$. This means $\vec{A} - \vec{C} = \lambda \vec{B}$ for some scalar $\lambda$.
2. Distributive Property of Dot Product: $\vec{X} \cdot (\vec{Y} + \vec{Z}) = \vec{X} \cdot \vec{Y} + \vec{X} \cdot \vec{Z}$.
3. Scalar Multiplication with Dot Product: $(\lambda \vec{X}) \cdot \vec{Y} = \lambda (\vec{X} \cdot \vec{Y})$.
4. Dot Product of Vectors: If $\vec{U} = u_1 \hat{i} + u_2 \hat{j} + u_3 \hat{k}$ and $\vec{V} = v_1 \hat{i} + v_2 \hat{j} + v_3 \hat{k}$, then $\vec{U} \cdot \vec{V} = u_1 v_1 + u_2 v_2 + u_3 v_3$.

Step-by-Step Solution

We are given three vectors $\vec{a}=3 \hat{i}+\hat{j}-2 \hat{k}$, $\vec{b}=4 \hat{i}+\hat{j}+7 \hat{k}$, and $\vec{c}=\hat{i}-3 \hat{j}+4 \hat{k}$. We are also given two conditions for an unknown vector $\vec{p}$:

1. $\vec{p} \times \vec{b}=\vec{c} \times \vec{b}$
2. $\vec{p} \cdot \vec{a}=0$

Our goal is to find the value of $\vec{p} \cdot(\hat{i}-\hat{j}-\hat{k})$.

Step 1: Use the cross product condition to establish a relationship for $\vec{p}$.

The first condition is $\vec{p} \times \vec{b}=\vec{c} \times \vec{b}$. Rearranging this equation, we get: $\vec{p} \times \vec{b} - \vec{c} \times \vec{b} = \vec{0}$ Using the distributive property of the cross product, we can write: $(\vec{p} - \vec{c}) \times \vec{b} = \vec{0}$ This implies that the vector $(\vec{p} - \vec{c})$ is parallel to the vector $\vec{b}$. Therefore, we can express $(\vec{p} - \vec{c})$ as a scalar multiple of $\vec{b}$: $\vec{p} - \vec{c} = \lambda \vec{b}$ where $\lambda$ is a scalar. From this, we can express $\vec{p}$ as: $\vec{p} = \vec{c} + \lambda \vec{b} \quad (*)$

Step 2: Use the dot product condition to find the scalar $\lambda$.

The second condition is $\vec{p} \cdot \vec{a} = 0$. Substitute the expression for $\vec{p}$ from equation $(*)$ into this condition: $(\vec{c} + \lambda \vec{b}) \cdot \vec{a} = 0$ Using the distributive property of the dot product, we expand this to: $\vec{c} \cdot \vec{a} + (\lambda \vec{b}) \cdot \vec{a} = 0$ $\vec{c} \cdot \vec{a} + \lambda (\vec{b} \cdot \vec{a}) = 0$

Now, we calculate the required dot products: $\vec{c} \cdot \vec{a} = (\hat{i}-3 \hat{j}+4 \hat{k}) \cdot (3 \hat{i}+\hat{j}-2 \hat{k})$ $\vec{c} \cdot \vec{a} = (1)(3) + (-3)(1) + (4)(-2) = 3 - 3 - 8 = -8$

$\vec{b} \cdot \vec{a} = (4 \hat{i}+\hat{j}+7 \hat{k}) \cdot (3 \hat{i}+\hat{j}-2 \hat{k})$ $\vec{b} \cdot \vec{a} = (4)(3) + (1)(1) + (7)(-2) = 12 + 1 - 14 = -1$

Substitute these values back into the equation $\vec{c} \cdot \vec{a} + \lambda (\vec{b} \cdot \vec{a}) = 0$: $-8 + \lambda (-1) = 0$ $-8 - \lambda = 0$ $\lambda = -8$

Step 3: Determine the vector $\vec{p}$.

Now that we have found $\lambda = -8$, we can substitute this value back into the expression for $\vec{p}$ from equation $(*)$: $\vec{p} = \vec{c} + (-8) \vec{b}$ $\vec{p} = (\hat{i}-3 \hat{j}+4 \hat{k}) - 8(4 \hat{i}+\hat{j}+7 \hat{k})$ $\vec{p} = (\hat{i}-3 \hat{j}+4 \hat{k}) - (32 \hat{i}+8 \hat{j}+56 \hat{k})$ $\vec{p} = (1 - 32) \hat{i} + (-3 - 8) \hat{j} + (4 - 56) \hat{k}$ $\vec{p} = -31 \hat{i} - 11 \hat{j} - 52 \hat{k}$

Step 4: Calculate the final dot product $\vec{p} \cdot(\hat{i}-\hat{j}-\hat{k})$.

We need to compute the dot product of $\vec{p}$ with the vector $(\hat{i}-\hat{j}-\hat{k})$. Let $\vec{q} = \hat{i}-\hat{j}-\hat{k}$. $\vec{p} \cdot \vec{q} = (-31 \hat{i} - 11 \hat{j} - 52 \hat{k}) \cdot (\hat{i}-\hat{j}-\hat{k})$ Using the component form of the dot product: $\vec{p} \cdot \vec{q} = (-31)(1) + (-11)(-1) + (-52)(-1)$ $\vec{p} \cdot \vec{q} = -31 + 11 + 52$ $\vec{p} \cdot \vec{q} = -31 + 63$ $\vec{p} \cdot \vec{q} = 32$

Common Mistakes & Tips

• Do not assume $\vec{p} = \vec{c}$ directly from $\vec{p} \times \vec{b}=\vec{c} \times \vec{b}$. This only implies that $(\vec{p} - \vec{c})$ is parallel to $\vec{b}$.
• Carefully compute the dot products to avoid arithmetic errors, as these values are crucial for finding the scalar $\lambda$.
• Remember to correctly distribute the scalar multiplication when calculating the components of $\vec{p}$.

Summary

The problem requires using the properties of vector cross and dot products to find an unknown vector $\vec{p}$. The condition $\vec{p} \times \vec{b}=\vec{c} \times \vec{b}$ allows us to express $\vec{p}$ in terms of $\vec{c}$ and $\vec{b}$ with an unknown scalar $\lambda$. The second condition, $\vec{p} \cdot \vec{a}=0$, is then used to form an equation that allows us to solve for $\lambda$. Once $\lambda$ is found, $\vec{p}$ is fully determined, and the required dot product can be calculated. The calculated value is 32.

The final answer is $\boxed{32}$.