Question

show that a.b = axbx + ayby + azbz, if a = ax i + ay j + az k and b =bx i + by j + bz k​

Answer 1

a.b=(axi+ayj+azk).(bxi+byj+bzk)

a.b=axbx+ayby+azbz

Answer 2

Given that,

$a=a_xi+a_yj+a_zk\\\\b=b_xi+b_yj+b_zk$

To proof,

$a{\cdot}b=a_xb_x+a_yb_y+a_zb_z$

Solution,

We know that the dot product of two vectors give a scalar number. So,

$a{\cdot} b=(a_xi+a_yj+a_zk){\cdot} (b_xi+b_yj+b_zk)\\\\a{\cdot} b=a_xb_x(i\cdot} i)+a_yb_y(j\cdot} j)+a_zb_z(k\cdot} k)$

Since, i.i = j.j = k.k = 1

So,

$a{\cdot} b=a_xb_x+a_yb_y+a_zb_z$

Hence, proved

Learn more,

Vectors

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