Let a^{\to} = 4i^{\^}+5j^{\^}-k^{\^} , b^{\to} = i^{\^}-4j^{\^}+5k^{\^} and c^{\to} = 3i^{\^}+j^{\^}-k^{\^}. Find a vector d^{\to} which is perpendicular to both c^{\to} and b^{\to} and d^{\to}.a^{\to} = 21.

CBSE Class 12 Math 2018 Solved Paper - Question 20

Question : 20 of 29

Marks: +1, -0

Let

a↖{→}

4i↖{\^}+5j↖{\^}-k↖{\^}

b↖{→}

i↖{\^}-4j↖{\^}+5k↖{\^}

and

c↖{→}

3i↖{\^}+j↖{\^}-k↖{\^}

. Find a vector

d↖{→}

which is perpendicular to both

c↖{→}

and

b↖{→}

and

d↖{→}.a↖{→}

= 21.

Solution:

Let

d↖{→}

xi↖{\^}+yj↖{\^}+zk↖{\^}

since

d↖{→}

is perpendicular to

c↖{→}

and

b↖{→}

, so their dot product is zero

d↖{→}.c↖{→}

= 0 and

d↖{→}.b↖{→}

= 0
3x + y - z = 0 ... (1)
x - 4y + 5z = 0 ... (2)
Also given that

d↖{→}.a↖{→}

= 21.
4x + 5y - z = 21 ... (3)
Solving all the equations simultaneously
x =

-1/3

, y =

{16}/3

, z =

{13}/3

d↖{→}

1/3

(-i↖{\^}+16j↖{\^}+13k↖{\^})

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