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
ANSWERED
asked 2020-10-21

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

Answers (1)

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,
\(b=c\)

0
 
Best answer

expert advice

Have a similar question?
We can deal with it in 3 hours
...