# Perpendicular vectorsFind a if the vectors u=3i+5j , v=ai-6j are perpendiculars .

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The vectors u = 3i + 5j and v = ai - 6j are perpendicular if their dot product is equal to zero. The dot product of the given vectors is 3*a + 5*(-6)

3*a + 5*(-6) = 0

=> 3a = 30

=> a = 10

The value of a should be 10 for u = 3i + 5j and v = ai - 6j to be perpendicular.

If 2 vectors are perpendicular, then the dot product is zero.

u*v = 0

The dot vector is:

u*v = |u|*|v|*cos (u,v)

Since u and v are perpendicular, then cos 90 = 0.

u*v = 0

u*v = (3i+5j)(ai-6j)

We'll remove the brackets:

u*v = 3ai^2 - 18ij + 5aij - 30j^2

i^2 = |i|*|i| cos 0

But |i| =1 and cos 0 = 1

i^2 = j^2 = 1

i*j = 0

u*v = 3a - 30

We'll set u*v = 0

3a - 30 = 0

We'll divide by 3:

a - 10 = 0

We'll add 10:

a = 10

**The vector v = 10i - 6j is perpendicular to u = 3i+5j, for a = 10.**