The perpendicular distance, of the line 2x−1=−1y+2=2z+3 from the point P(2,−10,1), is:
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Step-by-Step Solution
Step 1: Identify line equation and point
First, we identify the given line in symmetric form. From the equation, we can extract a point P0 on the line and the direction vector d of the line. We are also given the point P from which we need to find the perpendicular distance.
Step 2: Find vector from point on line to given point
Next, we calculate the vector P0P by subtracting the coordinates of the point P0 on the line from the coordinates of the given point P. This vector connects a point on the line to the external point.
Step 3: Calculate cross product magnitude
To find the perpendicular distance, we use the formula involving the cross product. We compute the cross product of the vector P0P and the direction vector d. Then, we find the magnitude of this resulting cross product vector.
Step 4: Calculate magnitude of direction vector
We also need the magnitude of the direction vector d for the distance formula. We calculate this by taking the square root of the sum of the squares of its components.
Step 5: Calculate perpendicular distance
Finally, we apply the formula for the perpendicular distance from a point to a line: D=∣d∣∣P0P×d∣. We substitute the calculated magnitudes to find the final distance.