Distance from P = (1, 2, 3) to the line through the origin in the direction v = (1, −2, 2) equals which value exactly?

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Multiple Choice

Distance from P = (1, 2, 3) to the line through the origin in the direction v = (1, −2, 2) equals which value exactly?

To find the distance from a point to a line through the origin in a given direction, use the part of the point’s position vector that is perpendicular to the line. The closest point on the line is the projection of P onto the line, and the distance is the length of P minus that projection. Compute the projection of P onto v: P·v = 1·1 + 2·(-2) + 3·2 = 3 v·v = 1^2 + (-2)^2 + 2^2 = 9 Projection = (P·v)/(v·v) · v = (3/9) v = (1/3)(1, -2, 2) = (1/3, -2/3, 2/3) Difference P − projection = (1 − 1/3, 2 − (−2/3), 3 − 2/3) = (2/3, 8/3, 7/3) Distance = ||P − projection|| = sqrt[(2/3)^2 + (8/3)^2 + (7/3)^2] = sqrt[(4 + 64 + 49)/9] = sqrt(117/9) = sqrt(13) The distance is sqrt(13).

To find the distance from a point to a line through the origin in a given direction, use the part of the point’s position vector that is perpendicular to the line. The closest point on the line is the projection of P onto the line, and the distance is the length of P minus that projection.

Compute the projection of P onto v:

P·v = 1·1 + 2·(-2) + 3·2 = 3

v·v = 1^2 + (-2)^2 + 2^2 = 9

Projection = (P·v)/(v·v) · v = (3/9) v = (1/3)(1, -2, 2) = (1/3, -2/3, 2/3)

Difference P − projection = (1 − 1/3, 2 − (−2/3), 3 − 2/3) = (2/3, 8/3, 7/3)

Distance = ||P − projection|| = sqrt[(2/3)^2 + (8/3)^2 + (7/3)^2] = sqrt[(4 + 64 + 49)/9] = sqrt(117/9) = sqrt(13)

The distance is sqrt(13).

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