If `veca=3hati-hatj+5hatk` and `vecb=hati+2hatj-3hatk` are given vectors. A vector `vecc` which is perpendicular to z-axis satisfying `vecc.veca=9` and `vecc.vecb=-4`. If inclination of `vecc` with x-axis and y-axis and y-axis is `alpha` and `beta` respectively, then which of the following is not true?

Options

A. `alpha gt (pi)/(4)`
B. `beta gt (pi)/(2)`
C. `alpha gt (pi)/(2)`
D. `beta lt (pi)/(2)`

Answer

To solve the problem step by step, we will follow the given conditions and equations systematically.

### Step 1: Define the vectors
Given:
- \( \vec{a} = 3\hat{i} - \hat{j} + 5\hat{k} \)
- \( \vec{b} = \hat{i} + 2\hat{j} - 3\hat{k} \)

Let \( \vec{c} = x\hat{i} + y\hat{j} + z\hat{k} \).

### Step 2: Determine the condition for \( \vec{c} \) being perpendicular to the z-axis
Since \( \vec{c} \) is perpendicular to the z-axis, we have:
- \( z = 0 \)

Thus, \( \vec{c} \) simplifies to:
- \( \vec{c} = x\hat{i} + y\hat{j} \)

### Step 3: Use the dot product conditions
We are given two conditions:
1. \( \vec{c} \cdot \vec{a} = 9 \)
2. \( \vec{c} \cdot \vec{b} = -4 \)

#### Condition 1:
Calculating \( \vec{c} \cdot \vec{a} \):
\[ \vec{c} \cdot \vec{a} = (x\hat{i} + y\hat{j}) \cdot (3\hat{i} - \hat{j} + 5\hat{k}) = 3x - y = 9 \]
This gives us our first equation:
\[ 3x - y = 9 \quad \text{(Equation 1)} \]

#### Condition 2:
Calculating \( \vec{c} \cdot \vec{b} \):
\[ \vec{c} \cdot \vec{b} = (x\hat{i} + y\hat{j}) \cdot (\hat{i} + 2\hat{j} - 3\hat{k}) = x + 2y = -4 \]
This gives us our second equation:
\[ x + 2y = -4 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations
We have the following system of equations:
1. \( 3x - y = 9 \)
2. \( x + 2y = -4 \)

From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 3x - 9 \]

Substituting this into Equation 2:
\[ x + 2(3x - 9) = -4 \]
\[ x + 6x - 18 = -4 \]
\[ 7x - 18 = -4 \]
\[ 7x = 14 \]
\[ x = 2 \]

Now substituting \( x = 2 \) back into Equation 1 to find \( y \):
\[ 3(2) - y = 9 \]
\[ 6 - y = 9 \]
\[ -y = 3 \quad \Rightarrow \quad y = -3 \]

Thus, we have:
\[ \vec{c} = 2\hat{i} - 3\hat{j} \]

### Step 5: Calculate angles \( \alpha \) and \( \beta \)
The angles \( \alpha \) and \( \beta \) are the inclinations of \( \vec{c} \) with the x-axis and y-axis, respectively.

#### Angle \( \alpha \):
Using the formula:
\[ \cos(\alpha) = \frac{\vec{c} \cdot \hat{i}}{|\vec{c}|} \]
Calculating \( |\vec{c}| \):
\[ |\vec{c}| = \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \]
Now, calculating \( \cos(\alpha) \):
\[ \vec{c} \cdot \hat{i} = 2 \quad \Rightarrow \quad \cos(\alpha) = \frac{2}{\sqrt{13}} \]
Thus,
\[ \alpha = \cos^{-1}\left(\frac{2}{\sqrt{13}}\right) \]

#### Angle \( \beta \):
Using the formula:
\[ \cos(\beta) = \frac{\vec{c} \cdot \hat{j}}{|\vec{c}|} \]
Calculating \( \cos(\beta) \):
\[ \vec{c} \cdot \hat{j} = -3 \quad \Rightarrow \quad \cos(\beta) = \frac{-3}{\sqrt{13}} \]
Thus,
\[ \beta = \cos^{-1}\left(\frac{-3}{\sqrt{13}}\right) \]

### Step 6: Analyze the angles
We can conclude:
- \( \alpha \) is an angle in the first quadrant (since \( \cos(\alpha) > 0 \)).
- \( \beta \) is an angle in the second quadrant (since \( \cos(\beta) < 0 \)).

### Step 7: Determine which statement is not true
Based on the values of \( \alpha \) and \( \beta \):
- \( \alpha \) is greater than \( 45^\circ \) (true).
- \( \beta \) is greater than \( 90^\circ \) (true).

Thus, we can conclude which of the statements is not true based on the provided options.