Find the angle between the vector  from Mathematics Vector Alg

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 Multiple Choice QuestionsShort Answer Type

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311.

Find the angle between the vector straight i with hat on top minus straight j with hat on top space and space straight j with hat on top space minus space straight k with hat on top.


 Let space straight a with rightwards arrow on top space equals space straight i with hat on top minus stack straight j semicolon with hat on top space space straight b with rightwards arrow on top space equals space straight j with hat on top space minus space straight k with hat on top
straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space left parenthesis straight i with hat on top minus straight j with hat on top right parenthesis space left parenthesis straight j with hat on top space minus space straight k with hat on top right parenthesis space equals space 1 cross times 0 space plus left parenthesis negative 1 right parenthesis cross times 1 space plus 0 space cross times left parenthesis negative 1 right parenthesis space equals space minus 1
open vertical bar straight a with rightwards arrow on top close vertical bar space equals space square root of 1 squared plus left parenthesis negative 1 right parenthesis squared plus 0 squared end root space equals square root of 2
open vertical bar straight b with rightwards arrow on top close vertical bar space equals square root of 0 squared plus 1 squared plus left parenthesis negative 1 right parenthesis squared end root space equals square root of 2
We space know space that space straight a with rightwards arrow on top. straight b with rightwards arrow on top space equals open vertical bar straight a with rightwards arrow on top close vertical bar open vertical bar straight b with rightwards arrow on top close vertical bar cos space straight theta
Thus comma space cosθ space equals space fraction numerator straight a with rightwards arrow on top. straight b with rightwards arrow on top over denominator open vertical bar straight a with rightwards arrow on top close vertical bar. open vertical bar straight b with rightwards arrow on top close vertical bar end fraction space equals space fraction numerator negative 1 over denominator square root of 2 cross times square root of 2 end fraction equals fraction numerator negative 1 over denominator 2 end fraction
rightwards double arrow space space space cosθ space equals space cos 120 degree
rightwards double arrow space space space space straight theta space equals space 120 degree

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312.

Find the value of 'p' for which the vectors 3 straight i with hat on top plus 2 straight j with hat on top plus 9 straight k with hat on top space and space straight i with hat on top minus 2 straight p straight j with hat on top plus 3 straight k with hat on top are parallel.

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313.

If the cartesian equations of a line are fraction numerator 3 minus straight x over denominator 5 end fraction equals fraction numerator straight y plus 4 over denominator 7 end fraction equals fraction numerator 2 straight z minus 6 over denominator 4 end fraction comma write the vector equation for the line. 

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314. If space straight a with rightwards arrow on top space and space straight b with rightwards arrow on top space are space perpendicular space vectors comma space open vertical bar straight a with rightwards arrow on top plus straight b with rightwards arrow on top close vertical bar space equals space 13 space and space open vertical bar straight a with rightwards arrow on top close vertical bar space equals 5 space and space find space the space value space of space open vertical bar straight b with rightwards arrow on top close vertical bar.
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315.

Show that the four points A, B, C and D with position vectors
4 straight i with hat on top plus 5 straight j with hat on top plus straight k with hat on top comma negative straight j with hat on top minus straight k with hat on top comma space 3 straight i with hat on top plus 9 straight j with hat on top plus 4 straight k with hat on top space and space 4 left parenthesis negative straight i with hat on top plus straight j with hat on top plus straight k with hat on top right parenthesis respectively are coplanar.

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316.

The scalar product of the vector straight a with rightwards arrow on top equals space straight i with hat on top plus straight j with hat on top plus straight k with hat on top with a unit vector along the sum of vectors straight b with rightwards arrow on top equals 2 straight i with hat on top plus 4 straight j with hat on top minus 5 straight k with hat on top space space and space straight c with rightwards arrow on top space equals space straight lambda straight i with hat on top plus 2 straight j with hat on top plus 3 straight k with hat on top is equal to one. Find the value of straight lambda and hence find the unit vector along straight b with rightwards arrow on top plus straight c with rightwards arrow on top.

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 Multiple Choice QuestionsLong Answer Type

317.

Find the distance of the point (2, 12, 5) from the point of intersection of the line 
straight r with rightwards arrow on top equals 2 straight i with hat on top minus 4 straight j with hat on top plus 2 straight k with hat on top plus straight lambda open parentheses 3 straight i with hat on top plus 4 straight j with hat on top plus 2 straight k with hat on top close parentheses space and space the space plane space straight r with rightwards arrow on top. open parentheses straight i with hat on top minus 2 straight j with hat on top plus straight k with hat on top close parentheses equals 0.

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 Multiple Choice QuestionsShort Answer Type

318.

If a unit vector straight a with rightwards arrow on top space equals space straight x straight i with hat on top plus 2 straight j with hat on top minus straight z straight k with hat on top and straight b with rightwards arrow on top space equals space 3 straight i with hat on top minus straight y straight j with hat on top plus straight k with hat on top are two equal vectors, then write the value of x+y+z.

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319.

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

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320.

If straight a with rightwards arrow on top space and space straight b with rightwards arrow on top are two vectors such that open vertical bar straight a with rightwards arrow on top plus straight b with rightwards arrow on top close vertical bar equals open vertical bar straight a with rightwards arrow on top close vertical bar comma then prove that vector 2 straight a with rightwards arrow on top plus straight b with rightwards arrow on top is perpendicular to vector straight b with rightwards arrow on top.

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