MHB It is given that the lines intersect. Find the value of a.

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The discussion focuses on finding the value of 'a' for two intersecting lines defined by their parametric equations. To determine 'a', the i, j, and k components of both lines must be set equal, leading to a system of equations. By solving the equations for the parameters r and t, the values can be substituted into the k equation to find 'a'. The intersection condition implies that there will be three equations involving two variables, allowing for a solution. Ultimately, the value of 'a' can be derived through this method.
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The equation of the lines l1 and l2 are are r=(4+t)i + (a+3t)j + (2-3t)k and r=(1-2s)i + (1-s)j + (1+s)k respectively, where t and s are real parameters. It is given that the lines intersect. Find the value of a.
 
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Punch said:
The equation of the lines l1 and l2 are are r=(4+t)i + (a+3t)j + (2-3t)k and r=(1-2s)i + (1-s)j + (1+s)k respectively, where t and s are real parameters. It is given that the lines intersect. Find the value of a.

Since the lines intersect, there must be some point where the i, j and k components are all equal. So set them equal to each other and try to solve the system.
 
Notice that you will have three equations in only two variables. Solve the i and j equations for r and t, then put those values into the k equation to determine a.
 
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