 Hello and welcome to the session. In this session we discuss the following question that says show that the lines vector r equal to a minus t i cap plus a j cap plus a plus t k cap plus lambda into alpha minus delta i cap plus alpha j cap plus alpha plus delta k cap and vector r equal to b minus c i cap plus b j cap plus b plus c k cap plus mu into beta minus gamma i cap plus beta j cap plus beta plus gamma k cap. Our co-planer also find the equation of the plane containing both these lines. Consider the two lines vector r equal to vector a1 plus lambda into vector b1 and vector r equal to vector a2 plus mu into vector b2. Now these two lines are co-planer if and only if we have vector a2 minus vector a1 dot vector b1 cross vector b2 is equal to 0 and we have the equation of the plane containing both the lines is given by vector r minus vector a1 this whole dot vector b1 cross vector b2 equal to 0 or vector r minus vector a2 this whole dot vector b1 cross vector b2 is equal to 0. Now this is the key idea that we use for this question. Now we proceed with the solution we are given the equations of the lines as vector r equal to a minus d i cap plus a j cap plus a plus d k cap plus lambda into alpha minus delta i cap plus alpha j cap plus alpha plus delta k cap. The other equation of the line is vector r equal to b minus c i cap plus b j cap plus b plus c k cap plus mu into beta minus gamma i cap plus beta j cap plus beta plus gamma k cap. So we take this as equation one and this equation as equation two. So these are the two lines we have to show that these two lines are co-planar. So from equation one we have vector a1 is equal to this that is a minus d i cap plus a j cap plus a plus d k cap then vector b1 is equal to this that is alpha minus delta i cap plus alpha j cap plus alpha plus delta k cap. Now from equation two we have vector a2 is equal to this that is b minus c i cap plus b j cap plus b plus c k cap and vector b2 is equal to this that is beta minus gamma i cap plus beta j cap plus beta plus gamma k cap. Now we have the condition for the two lines to be co-planar. This is the condition that is condition for two lines to be co-planar is vector a2 minus vector a1 this whole dot vector b1 cross vector b2 is equal to 0. First of all let's find out vector a2 minus vector a1 this is equal to b minus c into i cap plus b j cap plus b plus c into k cap. This is vector a2 minus vector a1 which is a minus d i cap plus a j cap plus a plus d into k cap. As we get vector a2 minus vector a1 is equal to b minus c minus a plus d into i cap plus b minus a into j cap plus b plus c minus a minus d into k cap this is vector a2 minus vector a1. Next we find vector b1 cross vector b2 this is equal to determinant i cap j cap k cap. Now we write the scalar components of vector b1 which are alpha minus delta alpha and alpha plus delta then in the third row we write the scalar components of vector b2 which are beta minus gamma beta and beta plus gamma. Solving further we get this is equal to i cap into alpha beta plus alpha gamma minus alpha beta minus beta delta minus j cap into alpha beta plus alpha gamma minus delta beta minus delta gamma minus alpha beta plus alpha gamma minus delta beta plus delta gamma plus k cap into alpha beta minus delta beta minus alpha beta plus alpha gamma. So here alpha beta cancels with minus alpha beta then here also alpha beta cancels with minus alpha beta then delta gamma cancels with minus delta gamma here alpha beta cancels with minus alpha beta. So we get this is equal to i cap into alpha gamma minus delta beta minus j cap into 2 alpha gamma minus 2 delta beta plus k cap into alpha gamma minus delta beta or you can say this is equal to alpha gamma minus delta beta this whole into i cap minus 2 j cap plus k cap this is vector b1 cross vector b2. Now let's find out what is vector a2 minus vector a1 this whole dot vector b1 cross vector b2. If this is equal to 0 then we can say that the two lines are and so this is equal to this is vector a2 minus vector a1 so we write here this is equal to b minus c minus a plus d into i cap plus b minus a into j cap plus b plus c minus a minus d into k cap this is vector a2 minus vector a1 this whole dot vector b1 cross vector b2 which is alpha gamma minus delta beta into i cap minus 2 j cap plus k cap. So we find the dot product of these two so this is equal to alpha gamma minus delta beta into b minus c minus a plus d minus 2b plus 2a plus b plus c minus a minus d here we have this 2b cancels with minus 2b c cancels with minus c then d cancels with minus d 2a cancels with minus 2a so we get this is equal to alpha gamma minus delta beta into 0 which would be equal to 0. So therefore we get vector a2 minus vector a1 this whole dot vector b1 cross vector b2 is equal to 0 hence we say that the two lines 1 and 2 are coplanar. Now the equation of the plane containing the lines 1 and 2 is given by vector r minus vector a1 this whole dot vector b1 cross vector b2 equal to 0 as given in the key idea we can have either of the two equations. So we will find out this equation now putting the values for vector a1 and vector b1 cross vector b2 we get vector r minus a minus d i cap plus aj cap plus a plus d k cap and this whole dot vector b1 cross vector b2 which is alpha gamma minus beta delta into i cap minus 2j cap plus k cap is equal to 0. So further we get alpha gamma minus beta delta this whole into vector r dot i cap minus 2j cap plus k cap minus a minus d i cap plus aj cap plus a plus d k cap this whole dot i cap minus 2j cap plus k cap and this is equal to 0. So further we get vector r dot i cap minus 2j cap plus k cap is equal to a minus d minus 2a plus a plus d that is vector r dot i cap minus 2j cap plus k cap is equal to a minus d minus 2a plus a plus d now here minus 2a cancels with 2a d cancels with minus d and so we get vector r dot i cap minus 2j cap plus k cap is equal to 0. So this is the required equation of the plane containing both the lines. So this completes the session, hope you have understood the solution of this question.