The intercepts made by the plane `vecr (2 hat(i) - 3 hat(j) + 4 hat(k) ) + 12 = 0` on the coordinate axes are

Options

A. `6,-4,3`
B. `4,-6,3`
C. `-6,4,-3`
D. `3,-4,6`

Answer

To find the intercepts made by the plane given by the equation \( \vec{r} \cdot (2 \hat{i} - 3 \hat{j} + 4 \hat{k}) + 12 = 0 \) on the coordinate axes, we can follow these steps:

### Step 1: Rewrite the Plane Equation
The given equation of the plane can be rewritten in the standard form. The dot product can be expanded as follows:
\[ \vec{r} \cdot (2 \hat{i} - 3 \hat{j} + 4 \hat{k}) + 12 = 0 \]
This implies:
\[ 2x - 3y + 4z + 12 = 0 \]

### Step 2: Rearranging the Equation
Rearranging the equation gives:
\[ 2x - 3y + 4z = -12 \]

### Step 3: Finding the Intercepts
To find the intercepts on the coordinate axes, we set two of the variables to zero and solve for the third.

1. **X-intercept**: Set \( y = 0 \) and \( z = 0 \):
\[ 2x = -12 \implies x = -6 \]
So, the x-intercept is \( (-6, 0, 0) \).

2. **Y-intercept**: Set \( x = 0 \) and \( z = 0 \):
\[ -3y = -12 \implies y = 4 \]
So, the y-intercept is \( (0, 4, 0) \).

3. **Z-intercept**: Set \( x = 0 \) and \( y = 0 \):
\[ 4z = -12 \implies z = -3 \]
So, the z-intercept is \( (0, 0, -3) \).

### Step 4: Summary of Intercepts
The intercepts made by the plane on the coordinate axes are:
- X-intercept: \( -6 \)
- Y-intercept: \( 4 \)
- Z-intercept: \( -3 \)

Thus, the intercepts are \( (-6, 4, -3) \).

### Final Answer
The intercepts made by the plane on the coordinate axes are \( (-6, 4, -3) \).

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