https://file.aiquickdraw.com/mathbot/file/1721908472393vev56y26.jpg

To find the perpendicular distance from the origin to the plane described by the equation:

\[
\vec{r} \cdot (2\hat{i} + 3\hat{j} - \hat{k}) = 5
\]

We can use the formula for the distance from a point to a plane. The general equation of a plane is given by:

\[
Ax + By + Cz = D
\]

The distance \( d \) from a point \((x_1, y_1, z_1)\) to the plane \(Ax + By + Cz = D\) is given by:

\[
d = \frac{|Ax_1 + By_1 + Cz_1 - D|}{\sqrt{A^2 + B^2 + C^2}}
\]

In our case, the point is the origin \((0, 0, 0)\), so the equation simplifies to:

\[
d = \frac{|D|}{\sqrt{A^2 + B^2 + C^2}}
\]

Here, \(A = 2\), \(B = 3\), \(C = -1\), and \(D = 5\). Plugging these values into the formula, we get:

\[
d = \frac{|5|}{\sqrt{2^2 + 3^2 + (-1)^2}} = \frac{5}{\sqrt{4 + 9 + 1}} = \frac{5}{\sqrt{14}}
\]

So, the perpendicular distance of the plane from the origin is:

\[
\frac{5}{\sqrt{14}} \text{ units}
\]

Thus, the correct answer is option (a):

\[
\frac{5}{\sqrt{14}} \text{ units}
\]

Do you want more details or have any questions?

Here are five more questions for you to consider:
1. How do you find the equation of a plane given three points in space?
2. What is the method to find the shortest distance between two skew lines?
3. How do you find the angle between two planes?
4. What is the distance formula between a point and a line in three-dimensional space?
5. How do you determine if a point lies on a plane given its equation?

**Tip:** Remember, when dealing with vector equations of planes, converting to the Cartesian form can often simplify distance and angle calculations.

Learn how to calculate the perpendicular distance from the origin to the plane described by the equation 2i + 3j - k = 5 using the distance formula. Understand the vector equation of a plane and its application in geometry. Suitable for advanced high school students and anyone interested in plane geometry and vector equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation