Find the shortest distance between the lines\(\overrightarrow r=(\hat i+2\hat j+\hat k)+\lambda(\hat i-\hat j+\hat k)\) and\(\overrightarrow r=2\hat i-\hat j-\hat k+\mu\,(2\hat i+2\hat j+2\hat k)\)
An aeroplane can carry a maximum of 200 passengers. A profit of Rs 1000 is made on each executive class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class.However, at least 4 times as many passengers refer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. What is the maximum profit?
Find the area of the parallelogram whose adjacent sides are determined by the vector \(\vec{a}=\hat{i}-\hat{j}+3\hat{k}\) and \(\vec{b}=2\hat{i}-7\hat{j}+\hat{k}.\)
Find the vector equation of the line passing through the point (1, 2, -4)and perpendicular to the two lines: \(\frac {x-8}{3}=\frac {y+19}{-16}=\frac {z-10}{7}\) and \(\frac {x-15}{3}=\frac {y-29}{8} =\frac {z-5}{-5}\)
If l1,m1,n1 and l2,m2,n2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m1n2-m2n1,n1l2-n2l1,l1m2-l2m1.
Prove that if a plane has the intercepts a, b, c and is at a distance of P units from the origin, then \(\frac {1}{a^2}+\frac {1}{b^2}+\frac {1}{c^2} =\frac {1}{p^2}\).
Maximise Z=3x+4ySubject to the constrains:x+y≤4,x≥0,y≥0.
Distance between the two planes: 2x+3y+4z = 4 and 4x+6y+8z = 12 is
\(The\ planes: 2x-y+4z=5 \ and \ 5x-2.5y+10z=6\ are\)
Find the angle between the planes whose vector equations are
\(\overrightarrow r.(2\hat i+2\hat j-3\hat k)=5\) and \(\overrightarrow r.(3\hat i-3\hat j+5\hat k)=3\)
If either \(\vec{a}=0\) or \(\vec{b}=0\),then \(\vec{a}\times\vec{b}=0\). Is the converse true? Justify your answer with an example.
Find the equation of the plane through the line of intersection of the planesx+y+z=1 and 2x+3y+4z=5 which is perpendicular to the plane x-y+z= 0