Published on 26 Jan 2015

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ENGG 202 Tutorial 3 January 22/23, 2013

* question is from a past ENGG 205 exam

3D vectors

1*) The line CA is contained in the x-y plane. Point

B lies in the x-z plane. Point O is located directly

under point A. Express the unit vector directed along

the line AB in Cartesian vector format given that the

angle OBA (α) is 65o and angle xOB (β) is 40o.

ANS: uAB = 0.324i + -0.906j + 0.272k

Solution Strategy: First, recognize that we are asked

to find components for the vector directed from A to

B, not from B to A. Since we want components of a

unit vector then just take the magnitude of the vector

as 1. Using the angle α, and considering the right triangle AOB, determine 2 components, 1 that is

on line AO and is parallel to the y-axis (so it is the y component) and 1 that is parallel to line OB.

Now break the OB component into 2 further components using the angle β, and considering the

right triangle HOB. 1 component will be parallel to the x axis (x component) and 1 is parallel to

the z axis (z component).

2*) The airplane’s engines exert a total

thrust force T with a magnitude of 200 kN.

The angle between T and the y axis is 120o,

and the angle between T and the z axis is

130o. The x component of T is positive.

(a) What is the angle between T and the x

axis?

(b) Express T in Cartesian vector format.

ANS: θ = 54.5o

T = (116.1i + -100.0j + -128.6k) kN

Soln Strategy: the two angles shown are two of the coordinate angles (angle between T and the

y and z axes. The third angle can be found from the relationship: cos2θx + cos2θy + cos2θz =1.

(direction cosines are the components of a unit vector). Note that using this equation (which

involves squares) will not give you the correct sign of the direction cosine so you have to pay

attention to whether the x component is positive (angle should be less than 90) or negative (angle

should be between 90 and 180). With all the angles know the components are simply: Tx=

Tcosθx , Ty= Tcosθy , Tz= Tcosθz

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