Supose that \(\displaystyle{a}\ne{0}\)

a) If \(\displaystyle{a}\cdot{b}={a}\cdot{c}\), then

\(\displaystyle{a}\cdot{b}-{a}\cdot{c}={0}\)

\(\displaystyle{a}\cdot{\left({b}-{c}\right)}={0}\)

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 \(\displaystyle{a}\times{b}={a}\times{c}={0}\)

\(\displaystyle{a}\times{b}-{a}\times{c}={0}\)

\(\displaystyle{a}\times{\left({b}-{c}\right)}={0}\)

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,

\(b=c\)