Engineering Calculus Notes 69

# Engineering Calculus Notes 69 - 57 1.5 PLANES 6 Consider...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 57 1.5. PLANES 6. Consider the line ℓ in the plane deﬁned as the locus of the linear equation in the two rectangular coordinates x and y Ax + By = C. Deﬁne the vector − → → → N = A− + B − . ı → (a) Show that ℓ is the set of points P whose position vector − p satisﬁes →→ −− N · p = C. → (b) Show that if − 0 is the position vector of a speciﬁc point on the p → line, then ℓ is the set of points P whose position vector − p satisﬁes →→→ −−− N · ( p − p 0 ) = 0. → − (c) Show that N is perpendicular to ℓ. 7. Show that if ℓ is a line given by Ax + By = C then the distance from a point Q(x, y ) to ℓ is given by the formula dist(Q, ℓ) = |Ax + By − C | √ . A2 + B 2 (1.20) → − → p (Hint: Let N be the vector given in Exercise 6, and − 0 the position vector of any point P0 on ℓ. Show that −→ − → → →→ dist(Q, ℓ) = proj− P0 Q = proj− (− − − 0 ) , and interpret this in q p N N terms of A, B , C , x and y .) 1.5 Planes Equations of Planes We noted earlier that the locus of a “linear” equation in the three rectangular coordinates x, y and z Ax + By + Cz = D (1.21) ...
View Full Document

## This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

Ask a homework question - tutors are online