Question

# 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

Vectors

This is the quesetion. Suppose that a does not equal 0.
a. if $$\displaystyle{a}\cdot{b}={a}\cdot{c}$$, does it follow that $$b=c$$?
b. if $$\displaystyle{a}\times{b}={a}\times{c}$$, does it follow that $$b=c$$ ?
c. if $$\displaystyle{a}\cdot{b}={a}\cdot{c}$$ and $$\displaystyle{a}\times{b}={a}\times{c}$$, does it follow that $$b=c$$?
Either prove the assertion is true in general or show that it is false for a concret choice of vectors a, b, c

2020-10-22

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,
$$b=c$$