Question

If vector a, vector b, vector c are any three vectors, show that vector a×(vector b+ vector c)=vector a× vector b+ vector a× vector

c.

Answer 1

Hey

Now, vector a+vector b+vector c=0

Taking cross product with vector a

Hence, vector a × vector a + vector a × vector b + vector a × vector c =0 ….. (I)

Vector a × vector a= 0

Hence, Expression (I) becomes……

0+vector a × vector b + vector a × vector c = 0

Hence, vector a × vector b = vector c × vector a

As, vector a × vector b = vector c × vector a

Hence, vector a × vector b = vector c × vector a.

i hope its help you

Answer 2

We know that; Both the Dot product and Cross product follow the property of distributive law:

So;

LHS = A⃗ × (B⃗ +C⃗ )

= (A⃗ × B⃗ ) + (A⃗ × B⃗ ) = RHS

Proof:

Let d=a⃗×(b⃗+c⃗)−a⃗×b⃗−a⃗×c⃗

so it is required to prove that d=0:

d² = d⃗•d⃗

=d⃗•(a⃗×(b⃗+c⃗)−a⃗×b⃗−a⃗×c⃗)

=d⃗•(a⃗×(b⃗+c⃗))−d⃗•(a⃗×b⃗)−d⃗•(a⃗×c⃗)

=(d⃗×a⃗)•(b⃗+c⃗)−(d⃗×a⃗)•b⃗−(d⃗×a⃗)•c⃗

=(d⃗×a⃗)•(b⃗+c⃗)−(d⃗×a⃗)•(b⃗+c⃗)

=0

Therefore d=0, so a⃗×(b⃗+c⃗)=(a⃗×b⃗)+(a⃗×c⃗)