If a unit vector  makes angle  and an acute angle  with 

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 Multiple Choice QuestionsLong Answer Type

191. Show that the vectors 2 straight i with hat on top minus straight j with hat on top plus straight k with hat on top comma space straight b with rightwards arrow on top space equals space straight i with hat on top space minus space 3 straight j with hat on top space minus space 5 straight k with hat on top comma space straight c with rightwards arrow on top space equals space 3 straight i with hat on top minus 4 straight j with hat on top minus 4 straight k with hat on top form the vertices of a right angled triangle. 

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192. Prove that the vectors
straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top comma space space space space space straight b with rightwards arrow on top space equals space straight i with hat on top space minus 3 space straight j with hat on top space minus space 5 space straight k with hat on top comma space space straight c with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 4 space straight k with hat on top forms the sides of a right-angled triangle. Also, find the remaining angles of the triangle.
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 Multiple Choice QuestionsShort Answer Type

193. Show that the points whose position vectors are 
straight a with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space 4 space straight j with hat on top space plus space space 5 space straight k with hat on top comma space space straight c with rightwards arrow on top space equals straight i with hat on top space minus space straight j with hat on top form a right angled triangle.
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194.

If the vertices A, B, C of ∆ ABC have position vectors (1, 2. 3), (– 1, 0, 0), (0, 1, 2) respectively, what is the magnitude of ∠ABC?

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195.

Show that 1 over 7 left parenthesis 2 straight i with hat on top space plus space 3 straight j with hat on top space plus space 6 straight k with hat on top right parenthesis comma space space 1 over 7 left parenthesis 3 straight i with hat on top minus 6 straight j with hat on top plus 2 straight k with hat on top right parenthesis comma space space 1 over 7 left parenthesis 6 straight i with hat on top plus 2 straight j with hat on top minus 3 straight k with hat on top right parenthesis are mutually perpendicular unit vectors.

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 Multiple Choice QuestionsLong Answer Type

196. The scalar product of the vector straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top with a unit vector along the sum of vectors 2 straight i with hat on top space plus space 4 space straight j with hat on top space minus space 5 straight k with hat on top and straight lambda straight i with hat on top space plus space 2 straight j with hat on top space plus space space 3 straight k with hat on top is equal to 1. Find the value of  λ.
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197. If a unit vector straight a with rightwards arrow on top makes angle straight pi over 4 space with space straight i with hat on top comma space space straight pi over 3 space with space straight j with hat on top and an acute angle straight theta with straight k with hat on top, then find the component of straight a with rightwards arrow on top and the angle straight theta.


Let a1 , a, a3 be scalar components of straight a with rightwards arrow on top.
   space therefore space space space space straight a with rightwards arrow on top space equals space straight a subscript 1 space straight i with hat on top space plus space straight a subscript 2 space straight j with hat on top space plus space straight a subscript 3 space straight k with hat on top
 Since straight a with rightwards arrow on top makes an angle straight pi over 4 space with space straight i with hat on top
therefore space space cos space straight pi over 4 space equals space fraction numerator straight a with rightwards arrow on top. space straight i with hat on top over denominator open vertical bar straight a with rightwards arrow on top close vertical bar. space open vertical bar straight i with hat on top close vertical bar end fraction space space space space space rightwards double arrow space space space space fraction numerator 1 over denominator square root of 2 end fraction space equals space fraction numerator straight a subscript 1 over denominator left parenthesis 1 right parenthesis thin space left parenthesis 1 right parenthesis end fraction space space space space space space space space space space space space space open square brackets because space space space open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 1 close square brackets
rightwards double arrow space space space straight a subscript 1 space equals space fraction numerator 1 over denominator square root of 2 end fraction
Again, straight a with rightwards arrow on top makes an angle straight pi over 3 space with space straight j with hat on top
therefore space space cos space straight pi over 3 space equals space fraction numerator straight a with rightwards arrow on top. space straight i with hat on top over denominator open vertical bar straight a with rightwards arrow on top close vertical bar. space open vertical bar straight j with hat on top close vertical bar end fraction space space space space space rightwards double arrow space space space space 1 half space equals space fraction numerator straight a subscript 2 over denominator left parenthesis 1 right parenthesis thin space left parenthesis 1 right parenthesis space end fraction space space space space space rightwards double arrow space space space straight a subscript 2 space equals space 1 half
Since straight a with rightwards arrow on top is a unit vector
therefore space space open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 1 space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space open vertical bar straight a with rightwards arrow on top close vertical bar squared space equals space 1 space space space space space rightwards double arrow space space space straight a subscript 1 squared plus straight a subscript 2 squared plus straight a subscript 3 squared space equals space 1
therefore space space space 1 half plus 1 fourth plus straight a subscript 3 squared space equals space 1 space space space space space space rightwards double arrow space space space straight a subscript 3 squared space equals 1 fourth space space space space rightwards double arrow space straight a subscript 3 space equals space plus-or-minus 1 half
Since space straight theta space space is space angle space between space straight a with rightwards arrow on top space and space straight k with hat on top
therefore space space cos space straight theta space equals space fraction numerator straight a with rightwards arrow on top. space straight k with hat on top over denominator open vertical bar straight a with rightwards arrow on top close vertical bar. space open vertical bar straight k with hat on top close vertical bar end fraction space equals fraction numerator straight a subscript 3 over denominator left parenthesis 1 right parenthesis left parenthesis 1 right parenthesis end fraction space equals space plus-or-minus 1 half.

∴    = 60°, 120°
Rejecting straight theta = 120° as ө is given to be acute,
we have straight theta = 160°
Now scalar components of straight a with rightwards arrow on top are fraction numerator 1 over denominator square root of 2 end fraction comma space 1 half comma space plus-or-minus 1 half and vector component of straight a with rightwards arrow on top are fraction numerator 1 over denominator square root of 2 end fraction space straight i with hat on top comma space 1 half space straight j with hat on top space plus-or-minus 1 half straight k with hat on top.
Also   straight theta space equals space 60 degree.

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 Multiple Choice QuestionsShort Answer Type

198. If A, B, C have position vectors (0, 1, 1) (3, 1, 5), (0. 3, 3) respectively, then show that the ∆ABC is right angled at C.
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 Multiple Choice QuestionsLong Answer Type

199. If a unit vector straight a with rightwards arrow on top makes angles straight pi over 3 with straight i with hat on top comma  straight pi over 4 space with space straight j with hat on top and acute angle straight theta with straight k with hat on top, then find the components of straight a with rightwards arrow on top and the angle straight theta.
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200.

Let straight a with rightwards arrow on top space equals space 4 straight i with hat on top space plus space 5 straight j with hat on top space minus space straight k with hat on top comma space straight b with rightwards arrow on top space equals space straight i with hat on top minus 4 straight j with hat on top plus space 5 straight k with hat on top space and space straight c with rightwards arrow on top space equals 3 straight i with hat on top plus straight j with hat on top minus straight k with hat on top. space Find a vector straight d with rightwards arrow on top which is perpendicular to both straight a with rightwards arrow on top space and space straight b with rightwards arrow on top space and space straight d with rightwards arrow on top. space straight c with rightwards arrow on top space equals space 21.

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