Vector question especially part ii) Fig. 7 illustrates a house. All units are in
ID: 2985736 • Letter: V
Question
Vector question especially part ii)
Fig. 7 illustrates a house. All units are in metres. The coordinates of A, B, C and E are as shown. BD is horizontal and parallel to AE. Find the length AE. Find a vector equation of the line BD. Given that the length of BD is 15 metres, find the coordinates of D. Verify that the equation of the plane ABC is -3x + 4y + 5z = 30. Write down a vector normal to this plane. Show that the vector is normal to the plane ABDE. Hence find the equation of the plane ABDE. Find the angle between the planes ABC and ABDE.Explanation / Answer
line BD is parallel to AE. So, the direction cosines of the line BD will be same as the direction cosines of the line AE, which are (-15/(sqrt(15^2 + 20^2), 20/(sqrt(15^2 + 20^2), 0 )
= (-15/25, 20/25, 0) = (-3/5, 4/5 , 0)
Also the line BD passes through B(-1,-7,11)
So, the equation of the line would be ( x + 1 ) / (-3/5) = ( y + 7 ) / (4/5) = ( z - 11) / 0 = t
x = -1 -3t/5
y = -7 + 4t/5
z = 11
So, the vector equation of the line BD is
r(t) = xi + yj + zk
= (-1 - 3t/5)i + (-7 + 4t/5)j + 11k
unit vector in the direction of BD is (-3/5) i + (4/5) j
Length of BD is 15 meters, so multiply it by 15 to get -9i + 12j. Now adding this vector to the initial point B(-1,-7,11) we get the point D as (-10,5,11)
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