Direction ratios of two lines are $$4$$,k,$$5$$ and $$2$$,−$$1$$,$$2$$. If these lines include an angle$$π$$$$4$$, then k can be

Question

Direction ratios of two lines are $$4$$,k,$$5$$ and $$2$$,−$$1$$,$$2$$. If these lines include an angle$$π$$$$4$$, then k can be

Infinity Learn · Accepted Answer

If θ is the acute angle between two lines having direction ratios $$a1$$,$$b1$$,$$c1and$$ $$a2$$,$$b2$$,$$c2then$$ $$cos$$θ$$=a1a2+b1b2+c1c2a12+b12+c12$$⋅$$a22+b22+c22It$$ implies that $$cos$$$$π$$$$4=8$$−$$k+1016+25+k24+1+412=18$$−$$k41+k23Apply$$ square on both sides $$12=18$$−$$k2941+k2941+k2=2324+k2$$−$$36k7k2+369=648$$−$$72k7k2+72k$$−$$279=0Split$$ the middle term to get the solutions for the equation $$7k2+93k$$−$$21k$$−$$279=0k7k+93$$−$$37k+93=0k$$−$$37k+93=0Therefore$$, the possible value of  k is $$3$$

Direction ratios of two lines are $〈4, k, 5〉$ and $〈2, - 1, 2〉$ . If these lines include an angle $\frac{π}{4}$ , then $k$ can be

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Direction ratios of two lines are 4,k,5 and 2,−1,2. If these lines include an angleπ4, then k can be

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Direction ratios of two lines are $〈4, k, 5〉$ and $〈2, - 1, 2〉$ . If these lines include an angle $\frac{π}{4}$ , then $k$ can be