Let `vec(p) = 2hat(i) + 3hat(j) - ahat(k), vec(q) = bhat(i) + 5hat(j) - hat(k)` and `vec(r) = hat(i) + hat(j) + 3hat(k)`. If `vec(p),vec(q), vec(r)` are coplanar and `vec(p).vec(q)` = 20 then a and b have the values.

Options

A. –1, 3
B. 9, 7
C. 5, 5
D. 13, 9

Answer

To solve the problem step by step, we will follow the outlined process to find the values of \( a \) and \( b \) such that the vectors \( \vec{p}, \vec{q}, \vec{r} \) are coplanar and \( \vec{p} \cdot \vec{q} = 20 \).

### Step 1: Write down the vectors
Given:
\[ \vec{p} = 2\hat{i} + 3\hat{j} - a\hat{k} \]
\[ \vec{q} = b\hat{i} + 5\hat{j} - \hat{k} \]
\[ \vec{r} = \hat{i} + \hat{j} + 3\hat{k} \]

### Step 2: Use the coplanarity condition
Vectors \( \vec{p}, \vec{q}, \vec{r} \) are coplanar if the determinant of the matrix formed by their components is zero:
\[ \begin{vmatrix} 2 & 3 & -a \\ b & 5 & -1 \\ 1 & 1 & 3 \end{vmatrix} = 0 \]

### Step 3: Calculate the determinant
Expanding the determinant:
\[ = 2 \begin{vmatrix} 5 & -1 \\ 1 & 3 \end{vmatrix} - 3 \begin{vmatrix} b & -1 \\ 1 & 3 \end{vmatrix} - a \begin{vmatrix} b & 5 \\ 1 & 1 \end{vmatrix} \]
Calculating the minors:
\[ = 2(5 \cdot 3 - (-1) \cdot 1) - 3(b \cdot 3 - (-1) \cdot 1) - a(b \cdot 1 - 5 \cdot 1) \]
\[ = 2(15 + 1) - 3(3b + 1) - a(b - 5) \]
\[ = 32 - 9b - 3 - ab + 5a \]
Setting the determinant to zero:
\[ 29 - 9b + 5a - ab = 0 \quad \text{(Equation 1)} \]

### Step 4: Use the dot product condition
The dot product condition gives us:
\[ \vec{p} \cdot \vec{q} = 20 \]
Calculating the dot product:
\[ (2\hat{i} + 3\hat{j} - a\hat{k}) \cdot (b\hat{i} + 5\hat{j} - \hat{k}) = 2b + 15 + a = 20 \]
Rearranging gives:
\[ a + 2b = 5 \quad \text{(Equation 2)} \]

### Step 5: Substitute Equation 2 into Equation 1
From Equation 2, we can express \( a \):
\[ a = 5 - 2b \]
Substituting into Equation 1:
\[ 29 - 9b + 5(5 - 2b) - b(5 - 2b) = 0 \]
Expanding:
\[ 29 - 9b + 25 - 10b - 5b + 2b^2 = 0 \]
Combining like terms:
\[ 2b^2 - 24b + 54 = 0 \]
Dividing by 2:
\[ b^2 - 12b + 27 = 0 \]

### Step 6: Solve the quadratic equation
Using the quadratic formula:
\[ b = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \cdot 1 \cdot 27}}{2 \cdot 1} \]
\[ b = \frac{12 \pm \sqrt{144 - 108}}{2} \]
\[ b = \frac{12 \pm \sqrt{36}}{2} \]
\[ b = \frac{12 \pm 6}{2} \]
Thus, we have:
\[ b = 9 \quad \text{or} \quad b = 3 \]

### Step 7: Find corresponding values of \( a \)
1. If \( b = 9 \):
\[ a = 5 - 2(9) = 5 - 18 = -13 \]

2. If \( b = 3 \):
\[ a = 5 - 2(3) = 5 - 6 = -1 \]

### Final Values
Thus, the pairs \( (a, b) \) are:
1. \( ( -13, 9 ) \)
2. \( ( -1, 3 ) \)

### Conclusion
The values of \( a \) and \( b \) are \( a = -1 \) and \( b = 3 \).