7.5 Scalar and Vector Projections a Scalar Projection the Scalar Projection of the Vector a О Onto the Vector B Is a Scalar

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 SProj(ar onto b) =|| ar || cosθ =10cos120° = −5

r b) of the vector b onto the vector ar r r SProj(b onto ar) =|| b || cosθ =16cos120° = −8

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 SProj(ar onto ar) =|| ar || r r r r r b) If a ↑↓ b ( cosθ = −1), then SProj(a onto b) = − || a || r r c) If ar ⊥ b then SProj(ar onto b) = 0 b) the opposite vector −ar SProj(ar onto − ar) = − || ar ||

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 r r r a ⋅i r SProj(a onto i ) = r = 2 So, the scalar projection of the vector ar onto the vector b || i || can be written as: r r r r r r r a ⋅b b) of ar onto the vector i − j SProj(a onto b) =|| a || cosθ = r || b || r r r r r r a ⋅(i − j) (2,−3,4)⋅(1,−1,0) 5 Note: SProj(a onto i − j) = r r = = || i − j || 2 2 For a Cartesian (Rectangular) coordinate system, the scalar r components ax , a y , and az of a vector a = (a x , a y , a z ) are r r r r c) of ar onto the vector b = −i + 2 j + k r r the scalar projections of the vector a onto the unit vectors i , r r r r r r a ⋅b (2,−3,4) ⋅(−1,2,1) − 2 − 6 + 4 − 4 j , and k . SProj(a onto b) = r = = = || b || 6 1+ 4 +1 6 Proof:

r r (a ,a ,a )⋅(1,0,0) r r a ⋅i x y z r r SProj(a onto i ) = r = = ax d) of the unit vector i onto the vector a || i || 1 r r r r i ⋅a (1,0,0)⋅(2,−3,4) 2 SProj( i onto a) = r = = || a || 4 + 9 +16 29

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 = (a x , a y , a z ) are the vector projections of the r r r vector ar onto the unit vectors i , j , and k .

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 r r of the vector b but in the same plan containing the r (ar ⋅b)b (−6)(−1,0,3) (6,0,−18) ⎛ 3 − 9 ⎞ VProj(ar onto b) = = = = ,0, r r r 2 ⎜ ⎟ vectors a and b . || b || 1+ 0 + 9 10 ⎝ 5 5 ⎠

r b) the vector projection of the vector b onto the vector ar r r (b ⋅ar)ar (−6)(0,1,−2) (0,−6,12) ⎛ − 6 12 ⎞ VProj(b onto ar) = = = = ⎜0, , ⎟ || ar ||2 0 +1+ 4 5 ⎝ 5 5 ⎠

r r r Let a r be the vector component parallel to b . Then: c) the vector projection of the vector ar onto the unit vector k ||b r r r (ar ⋅ k )k (−2)(0,0,1) VProj(ar onto k ) = = = (0,0,−2) r r r r 2 r (a ⋅b)b || k || 1 ar r = VProj(ar onto b) = ||b r 2 || b || r r r r r d) the vector projection of the vector i onto the vector a Let a⊥b be the vector component perpendicular to b . r r r r r (i ⋅ a)a (0)(0,1,−2) r Then: VProj(i onto a) = = = (0,0,0) = 0 r r r || ar ||2 0 +1+ 4 r r r r (a ⋅b)b a r = a − a r = a − r ⊥b ||b || b ||2

Reading: Nelson Textbook, Pages 390-398 Homework: Nelson Textbook: Page 399 # 6, 11, 13, 14