This is the quesetion. Suppose that a does not equal 0. a. if a\cdot b=a\cdot c, does it follow that b=c? b. if a\times b=a\times c, does it follow th

Supose that ${\displaystyle a\ne 0}$
a) If ${\displaystyle a\cdot b=a\cdot c}$, then
${\displaystyle a\cdot b-a\cdot c=0}$
${\displaystyle a\cdot (b-c)=0}$
If the dot product of two vectors is zero then those vectors are perpendicular or orthogonal.
Thus, the vectors a and $(b-c)$ are perpendicular or orthogonal.
If the vectors have the same direction or one has zero length, then their dot product is zero.
Since ${\displaystyle a\ne 0}$, it follows that
${\displaystyle b-c\ne 0}$
${\displaystyle b\ne c}$
b) If ${\displaystyle a\times b=a\times c=0}$
${\displaystyle a\times b-a\times c=0}$
${\displaystyle a\times (b-c)=0}$
If the cross product of two vectors is zero then those vectors are parallel.
Thus, the vectors a and $(b-c)$ are parallel.
If the vectors have the same direction or one has zero length, then their cross product is zero.
Since ${\displaystyle a\ne 0}$, it follows that
${\displaystyle b-c\ne 0}$
${\displaystyle b\ne c}$
c) If ${\displaystyle \theta }$ is the angle between the vectors, then
${\displaystyle a\cdot b-\left|a\right|\left|b\right|\cos \theta }$
Let ${\displaystyle {\theta }_1}$ be the angle between the vectors a and b and ${\displaystyle {\theta }_1}$ be the angle between the vectors a and c.
So,
${\displaystyle a\cdot b=a\cdot c}$
${\displaystyle \left|a\right|\left|b\right|{\cos \theta }_1=\left|a\right|\left|c\right|{\cos \theta }_2}$
${\displaystyle \left|b\right|{\cos \theta }_1=\left|c\right|{\cos \theta }_2}$ (1)
Also,
${\displaystyle a\times b=a\times c}$
${\displaystyle \left|a\right|\left|b\right|{\sin \theta }_1=\left|a\right|\left|c\right|{\sin \theta }_2}$
${\displaystyle \left|b\right|{\sin \theta }_1=\left|c\right|{\sin \theta }_2}$ (2)
Dividing (1) by (2)
${\displaystyle \frac{{\sin \theta }_1}{{\cos \theta }_1}=\frac{{\sin \theta }_2}{{\cos \theta }_2}}$
${\displaystyle {\tan \theta }_1={\tan \theta }_2}$
${\displaystyle {\theta }_1={\theta }_2}$
Therefore,
$b=c$

(a) : if    a.b = a.c then     a.b - a.c = 0

which gives a.(b-c)=0

here a is perpendicular to b-c. so $b\ne c$

(b) : if $a\times b=a\times c$ this impliesthat a \times (b-c)

so a is parallel to b-c.

i.e. b may not be equal to c.

(c) : if a.b = a .c then in part (a) we have,    a is perpendicular to b - c.

also $a\times b=a\times c$

then    a isparallel to b -c.

now since a is not zero and parallel as well asperpendicular to b-c.

so we will have   b-c=0   i.e. b=c.