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euro94
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Find the direction vector of the line perpendicular to the line r=[-1,0,1] + t [2,3,-2] and passing through the point (-7,-9,7)
euro94 said:I tried finding the dot product of the line's direction vector with a vector from (-7,9,7). I tried (-1+2t, 3t, 1-2t) - (-7,-9,7) and I dotted it with (2,-3,2) and made it equal to zero, and then i got stuck :(
euro94 said:i mean i got a value of -27/14 for t, i had a miscalculation, but I am not sure what step to take next
euro94 said:okay :)
(-1+2t, 3t, 1-2t) - (-7,-9,7) = (6+2t, 3t+9, -6-2t)
dot product:
(6+2t, 3t+9, -6-2t) dot (2,3,-2) =0
12+4t+9t+27+12+4t =0
51+17t=0
-51/17=t
-3=t
r=[-1,0,1] + 3 [2,3,-2]
= [-7,-9,7]
A direction vector is a mathematical representation of the direction and magnitude of a line or a vector. It is typically expressed as a set of coordinates, such as (x,y,z) or i_{1}j_{1}k_{1}.
To find a direction vector when a line perpendicular and a point is given, you can use the cross product of the vector representing the perpendicular line and the vector connecting the given point to a point on the line. This will give you a direction vector that is perpendicular to the given line.
A direction vector is important in mathematics and physics as it allows us to determine the direction and orientation of a line or a vector. It is also used in various applications such as computer graphics, navigation, and optimization problems.
Yes, a direction vector can have negative components. This simply indicates the direction and magnitude of the vector in the opposite direction. For example, if a direction vector is (-2,3), it means that the vector is pointing in the negative x-direction with a magnitude of 2 and in the positive y-direction with a magnitude of 3.
Yes, a direction vector is unique for a given line or vector. This is because it represents the direction and orientation of the line or vector and is not affected by its magnitude. However, there may be multiple direction vectors that are perpendicular to a given line.