23 Maths -- Vectors

If A,B,C and D are any four points so that,vector AB.CD + vector BC.AD + vector CA.BD=0.

If A,B,C and D are any four points so that,vector AB.CD + vector BC.AD + vector CA.BD=0.

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More questions on Vectors

If | a"| = |b"| =|a"+b"| is given, Find |a'' - b''| [here " means vector 'Can you please tell me the concept behind this question?

ABCD is a parallelogram. P is the point of intersection of the diagonals and if O is any point show that OA⃗+OB⃗+OC⃗+OD⃗=4OP⃗

Given, ABCD is a parallelogram

P is point of intersection of the diagonals

O is any point

Let OA,OB,OC,OD and OP be drawn.

Then we have,

Now,

( For a parallelogram P bisects the two diagonal.)

If a⃗= (3,-1,-4), b⃗=(-2,4,-3) and c⃗=(-5,7,-1) findI) Unit vector along a⃗-2b⃗+c⃗ II) The direction cosines of the line represented by the vector a⃗-2b⃗+c⃗

Solution

Given vectors are

a= (3,-1,-4), b=(-2,4,-3) and c=(-5,7,-1)

a-2b+c = (3,-1,-4) -2(-2,4,-3)+(-5,7,-1)

= (3,-1,-4) -(-4,8,-6)+(-5,7,-1)

={3-(-4)+(-5) , -1-8+7, -4-(-6)+(-1)}

= (2,-2, 1)

I) For unit vector

| a-2b+c |= =

II) For direction cosines of the line represented by a-2b+c

=

Prove vectorically cos(A+B)=cosAcosB-sinAsinB

If 3i⃗+j⃗-k⃗ and xi⃗-4j⃗+4k⃗ are collinear vectors, then find the value of x.

Given vectors are

a= 3i+j-k

b= xi-4j+4k

since they are collinear, the relation of their coordinates is given as:

Taking 1^st and 2^nd ratio we get,