Question 2.135: For the points O, A, and B in Problem 2.134, use the cross p......
For the points O, A, and B in Problem 2.134, use the cross product to determine the length of the shortest straight line from point B to the straight line that passes through points O and A.
Points: O, A, B
Problem: 2.134
Step 1:
Find the cross product of vectors OA and OB to determine the vector C that is perpendicular to both OA and OB. This can be done by taking the determinant of the matrix formed by the components of OA and OB.
Step 2:
Calculate the components of vector C by evaluating the determinant.
Step 3:
Determine that vector C is perpendicular to both OA and OB.
Step 4:
Find a line that is perpendicular to the plane formed by vector C and OA. This line will be parallel to line BP on the diagram. This can be done by taking the cross product of vector C and OA.
Step 5:
Calculate the components of the cross product to find the direction of the line perpendicular to the plane.
Step 6:
Normalize the vector by dividing it by its magnitude to obtain the unit vector in the direction of C.
Step 7:
Determine the length of the projection of vector OB onto the line in the direction of C. This can be done by taking the dot product of OB and the unit vector in the direction of C.
Step 8:
Calculate the dot product to find the length of the projection.
Step 9:
The length of the projection, P, represents the distance from the point O to the line BP. In this case, the length is 6.90 meters.
Final Answer
\begin{aligned}r _{O A} & =6 i -2 j +3 k~( m ) \\r _{O B} & =4 i +4 j -4 k~( m ) \\r _{O A} \times r _{O B} & = C\end{aligned}
(C is ⊥ to both r_{OA}~and~r_{OB})
\begin{aligned}& C =\left|\begin{array}{rrr} i & j & k \\6 & -2 & 3 \\4 & 4 & -4\end{array}\right|=\begin{array}{r}(+8-12) i \\+(12+24) j \\+(24+8) k\end{array} \\\\& C =-4 i +36 j +32 k\end{aligned}C is ⊥ to both r_{OA}~and~r_{OB}. Any line ⊥ to the plane formed by C and r_{OA} will be parallel to the line BP on the diagram. C × r_{OA} is such a line. We then need to find the component of r_{OB} in this direction and compute its magnitude.
\begin{aligned}C \times r _{O A} & =\left|\begin{array}{rrr} i & j & k \\-4 & +36 & 32 \\6 & -2 & 3\end{array}\right| \\\\C & =172 i +204 j -208 k\end{aligned}The unit vector in the direction of C is
e _C=\frac{ C }{| C |}=0.508 i +0.603 j -0.614 k(The magnitude of C is 338.3)
We now want to find the length of the projection, P, of line OB in direction e_c.
\begin{aligned}P & = r _{O B} \cdot e _C \\& =(4 i +4 j -4 k ) \cdot e _C \\P & =6.90~m\end{aligned}Related Answered Questions
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