22UnderstandingDetermine the coordinates of the point that divides AB in the ratio 5:3 where A is (2, -1, 4) and B is (3, 1, 7)If AP:PB = m:n then
23UnderstandingConsider collinear points P, Q, and R. consider also a reference point O. Write OQ as a linear combination of OP and OR if:Q divides PR (internally) in the ratio 3:5Q divides PR (externally) in the ratio -2:7Since:Then:PQR35
24UnderstandingConsider collinear points P, Q, and R. consider also a reference point O. Write OQ as a linear combination of OP and OR if:Q divides PR (internally) in the ratio 3:5Q divides PR (externally) in the ratio -2:7Since:Then:7QPR-25
25UnderstandingDetermine the point Q that divides line segment PR in the ratio 5:8 if P is (1,5,2) and R is (-4,1,2).
26Inner (dot) product Defined for vectors: w v L w·cos(θ) is the scalar projection of w on vector v, which we will call Lwthe dot product can be understood geometrically as the product of the length of this projection and the length of v.vLThe dot product or scalar product allows us to determine a product the magnitude of one vector by the magnitude of the component of the other vector that’s parallel to the first. (The dot product was invented specifically for this purpose.)
27Dot product in coordinates yywAn interesting proofwyvvxwxvOx
28Proof of Geometric and Coordinate Definitions Equality From the Cosine Lawtherefore
33UnderstandingGiven that work is defined as (the dot product of force, f, and distance travelled, s. A crate on a ramp is hauled 8 m up the ramp under a constant force of 20 N applied at an angle of 300 to the ramp. Determine the work done.
34UnderstandingThe angle between two vectors u and v is 1100, if the magnitude of vector u is 12, then determine:The projection of vector u onto vector v
35The Cross ProductSome physical concepts (torque, angular momentum, magnetic force) require that we multiply the magnitude of one vector by the magnitude of the component of the other vector that is perpendicular to the first (recall in the dot product we used the parallel component).The cross product was invented for this specific purposeBsin(θ)The magnitude (length) of the cross productABθ
36Unlike the dot product, the cross product is a vector quantity. Where θ is the angle between and , and is a unit vector perpendicular to both and such that and form a right handed triangle.Take your right hand, and point all 4 fingers (not the thumb) in the direction of the first vector x. Next, rotate your arm or wrist so that you can curl your fingers in the direction of y. Then extend your thumb, which is perpendicular to both x and y. The thumb’s direction is the direction the cross product will point.
38UnderstandingFirst suppose u and v are collinear, then the angle, θ, between u and v is 00. Since sin(00)=0, then:Conversely suppose that then Since u and v are non-zero vectors, then sin(θ)=0, and hence θ=0 or θ=π.Therefore u and v are collinear.
40UnderstandingDetermine the area of the parallelogram below, using the cross product.830013
41UnderstandingState whether each expression has meaning. If not, explain why, if so, state whether it is a vector or a scalar quantity.scalarNon sensevectorNon sensevectorNon sensescalar
42UnderstandingA right-threaded bolt is tightened by applying a 50 N force to a 0.20 m wrench as shown in the diagram. Find the moment of the force about the centre of the bolt.The moment of the bolt is 9.4 Nm and is directed down
43Vector Product in Component Form Recall:But how do we obtain the answer in component form?The easiest technique is called the Sarrus’ Scheme
48Understanding Find a unit vector that is perpendicular to both: We need to apply the cross product to determine a vector perpendicular to both u and v. Then we divide by the length of this vector to make it a unit vector