The length of perpendicular from the origin to the plane :
`vec(r). (3 hati - 12 hatj - 4 hatk ) + 39 = 0 ` is ,

Options

A. 19
B. 3
C. 13
D. 12

Answer

To find the length of the perpendicular from the origin to the given plane, we can follow these steps:

1. **Identify the plane equation**: The equation of the plane is given as:
\[ \vec{r} \cdot (3 \hat{i} - 12 \hat{j} - 4 \hat{k}) + 39 = 0 \]
This can be rewritten in the standard form \(AX + BY + CZ + D = 0\), where \(A = 3\), \(B = -12\), \(C = -4\), and \(D = 39\).

2. **Identify the point**: The point from which we are measuring the distance is the origin, which can be represented as \(P(0, 0, 0)\).

3. **Use the distance formula**: The formula for the distance \(d\) from a point \((x_1, y_1, z_1)\) to the plane \(AX + BY + CZ + D = 0\) is given by:
\[ d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \]

4. **Substitute the values**: Here, substituting \(x_1 = 0\), \(y_1 = 0\), \(z_1 = 0\), \(A = 3\), \(B = -12\), \(C = -4\), and \(D = 39\):
\[ d = \frac{|3(0) - 12(0) - 4(0) + 39|}{\sqrt{3^2 + (-12)^2 + (-4)^2}} \]
This simplifies to:
\[ d = \frac{|39|}{\sqrt{9 + 144 + 16}} \]

5. **Calculate the denominator**: Now, calculate the denominator:
\[ \sqrt{9 + 144 + 16} = \sqrt{169} = 13 \]

6. **Final calculation of distance**: Therefore, the distance \(d\) becomes:
\[ d = \frac{39}{13} = 3 \]

Thus, the length of the perpendicular from the origin to the plane is **3 units**.