Calculus and Vectors – How to get an A+ 7.5 Scalar and Vector Projections A Scalar Projection Ex 1. Given two vectors with the magnitudes r r The scalar projection of the vector ar onto the vector b is a || ar ||=10 and || b ||=16 respectively, and the angle scalar defined as: between them equal to θ =120° , find the scalar r r SProj(ar onto b) =|| ar || cosθ where θ = ∠(ar,b) projection r a) of the vector ar onto the vector b r b) of the vector b onto the vector ar B Special Cases Ex 2. Find the scalar projection of the vector ar onto: r Consider two vectors ar and b . r r a) If ar ↑↑ b ( cosθ =1 ), then SProj(ar onto b) =|| ar || a) itself r r r r r b) If a ↑↓ b ( cosθ = −1), then SProj(a onto b) = − || a || r r r c) If ar ⊥ b then SProj(ar onto b) = 0 b) the opposite vector −a C Dot Product and Scalar Projection Ex 3. Given the vector ar = (2,−3,4) , find the scalar r Recall that the dot product of the vectors ar and b is defined projection: r as: a) of ar onto the unit vector i r r ar ⋅b =|| ar |||| b || cosθ r So, the scalar projection of the vector ar onto the vector b can be written as: r r r r r b) of a onto the vector i − j r r r a ⋅b SProj(a onto b) =|| a || cosθ = r || b || Note: For a Cartesian (Rectangular) coordinate system, the scalar r r r r c) of ar onto the vector b = −i + 2 j + k components a , a , and a of a vector ar = (a , a , a ) are x y z x y z r the scalar projections of the vector ar onto the unit vectors i , r r j , and k . r r Proof: d) of the unit vector i onto the vector a r r r r a ⋅i (ax ,a y ,az )⋅(1,0,0) SProj(a onto i ) = r = = ax || i || 1 D Vector Projection E Dot Product and Vector Projection r r The vector projection of the vector ar onto the vector b is a The vector projection of the vector a onto the vector r vector defined as: b can be written using the dot product as: r r r r r r r b r r (a ⋅b)b VProj(a onto b) =|| a || cosθ r VProj(a onto b) = r || b || || b ||2 Note: For a Cartesian (Rectangular) coordinate system, the r r r r r r vector components ax = axi , a y = a y j , and az = az k of r a vector a = (ax , a y , az ) are the vector projections of the r r r vector ar onto the unit vectors i , j , and k . 7.5 Scalar and Vector Projections ©2010 Iulia & Teodoru Gugoiu - Page 1 of 2 Calculus and Vectors – How to get an A+ r Ex 4. Given two vectors ar = (0,1,−2) and b = (−1,0,3) , find: Ex 5. Find an expression using the dot product of the vector components of the vector ar along to the vector r r r b and along to a direction perpendicular to the direction a) the vector projection of the vector a onto the vector b r of the vector b but in the same plan containing the r vectors ar and b . r b) the vector projection of the vector b onto the vector ar r c) the vector projection of the vector ar onto the unit vector k r d) the vector projection of the vector i onto the vector ar Reading: Nelson Textbook, Pages 390-398 Homework: Nelson Textbook: Page 399 # 6, 11, 13, 14 7.5 Scalar and Vector Projections ©2010 Iulia & Teodoru Gugoiu - Page 2 of 2 .