Tutorial 4 : scalar product and vector product - Mathématiques de M

Exercise 1 : using three ways to calculate a scalar product, calculate ... Exercise 3 : let P and Q be the planes of equation : P : x −4y +7 = 0; Q : x +2y −z +1 = 0. 1.
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INSA de ROUEN

2012/2013

Tutorial 4 : scalar product and vector product −→ −−→ Exercise 1 : using three ways to calculate a scalar product, calculate AE .DG where ABC DE F G is a cube of side a. − Exercise 2 : find a cartesian equation of the plane P passing by A(2; 1; −3) and for which → n = (1; 1; 2) is a normal vector. Exercise 3 : let P and Q be the planes of equation : P : x − 4y + 7 = 0 ; Q : x + 2y − z + 1 = 0. 1. Give a normal vector for each plane. Are they colinear ? Deduce that P and Q are secant. 2. Give a directing vector of their intersection line. Exercise 4 : let A(x A ; y A ; z A ) be a point and P a plane of equation ax + b y + c z + d = 0. We want to calculate the distance between the point A and the plane P . −−→ − − 1. Let H (x ; y ; z ) the normal projection of A on P . and Let → n a vector normal to P . Calculate AH .→ n using the coordiH

H

H

nates −−→ − −−→ − 2. Calculate AH .→ n considering the colinearity of AH and → n 3. Then deduce the distance AH . 4. Apply that result for A(1; 1; 1) and P : x + y + z = 1. Exercise 5 : using the vector product, find the equation of the plane passing by A(1; 0; 2), B (−1; 3; 0) and C (0; 2; 2). − → − → − → Exercise 6 : let (O, i , j , k ) be an orthonormal direct frame of the space. We consider A(1; 0; −1), B (2; 2; 3), C (3; 1; −2), D(−4; 2; 1), E (0; 0; 2) and M (x; y; z) 1. Show that A, B,C and D don’t belong to the same plane. Then calculate the volumen of the tetrahedron ABC D. −−→ −−→ 2. Calculate M B ∧ MC and deduce the surface of the triangle ABC and the distance between D and the plane (ABC ). → − 3. Let ∆ be the straight line passing by D and normal to (ABC ), V a vector normal to (ABC ) and H the normal projection of E on ∆. −−→ → − −−→ → − (a) Show that DE ∧ V = H E ∧ V . (b) Deduce the distance d between E and ∆. (c) Let K be the normal projection of D on (ABC ). Calculate D H and deduce the nature of the triangle E DK and give its surface. −−→ −−→ −−→ −−→ . Exercise 7 : considering A(−1; 0; −1), B (0, 1, 1) and C (1; 14; −1), calculate O A ∧ OB and O A.OB . Then deduce the angle AOB Finally, prove that O, A, B and C don’t belong to the same plane.