Chapter 6.1, Problem 54E

Solution

i.

To prove: The vector property using component form.

Given:

An expression .

Calculation:

Suppose that and .

Consider the left-hand side expression $u+v$ .

  

Apply the commutative property.

  

Conclusion:

Hence, It is proved that .

ii.

To prove: The vector property $\left(u+v\right)+w=u+\left(v+w\right)$ using component form.

Given:

An expression $\left(u+v\right)+w=u+\left(v+w\right)$ .

Calculation:

Suppose that and .

Substitute the assumed values in the left-hand expression $\left(u+v\right)+w$ .

  

Apply the associative rule of addition.

  

Conclusion:

Hence, it is proved that .

iii.

To prove: The vector property $u+0=u$ using component form.

Given:

An expression $u+0=u$ .

Calculation:

Suppose that and $\mathrm{<mn>0</mn>}=〈0,0〉$ .

Substitute the assumed values in the left-hand expression $u+0$ .

  

Conclusion:

Hence, it is proved that $u+0=u$ .

iv.

To prove: The vector property using component form.

Given:

An expression .

Calculation:

Suppose that .

Substitute the assumed values in the left-hand expression $u+\left(-u\right)$ .

  

Conclusion:

Hence, it is proved that .

v.

To prove: The vector property $\alpha \left(u+v\right)=\alpha u+\alpha v$ using component form.

Given:

An expression .

Calculation:

Suppose that and .

Substitute the assumed values in the left-hand expression $\alpha \left(u+v\right)$ .

  

It is understood that $a〈x,y〉=〈ax,ay〉$ . So,

  

Represent the expression as sum of two vectors.

  

Conclusion:

Hence, it is proved that .

vi.

To prove: The vector property using component form.

Given:

An expression .

Calculation:

Suppose that .

Substitute the assumed values in the left-hand expression .

  

It is understood that $a〈x,y〉=〈ax,ay〉$ . So,

  

Represent the expression as sum of two vectors.

  

Conclusion:

Hence, it is proved that .

vii.

To prove: The vector property using component form.

Given:

An expression .

Calculation:

Suppose that .

Substitute the assumed values in the left-hand expression .

  

Apply the associative rule of the multiplication.

  

Conclusion:

Hence, it is proved that .

viii.

To prove: The vector property using component form.

Given:

An expression .

Calculation:

Suppose that and $\mathrm{<mn>0</mn>}=〈0,0〉$ .

Substitute the assumed values in the left-hand expression $a0$ .

  $a\mathrm{<mn>0</mn>}=a〈0,0〉$

It is understood that $a〈x,y〉=〈ax,ay〉$ . So,

  $\begin{array}{c}a0=〈a\cdot 0,a\cdot 0〉\\ =〈0,0〉\\ =0\end{array}$

Consider another expression $0u=0$ .

  $\begin{array}{c}〈0,0〉\cdot 〈u_1,u_2〉=〈0\cdot u_1,0\cdot u_2〉\\ =〈0,0〉\\ =\mathrm{<mn>0</mn>}\end{array}$

Conclusion:

Hence, it is proved that $a\mathrm{<mn>0</mn>}=\mathrm{<mn>0</mn>}$ and $\mathrm{<mn>0</mn>}u=\mathrm{<mn>0</mn>}$ .

ix.

To prove: The vector property $\left(1\right)u=u,\left(-1\right)u=-u$ using component form.

Given:

An expression .

Calculation:

Suppose that .

Consider the expression .

Substitute the assumed values in left hand expression .

  

Consider another expression .

Substitute the assumed values in left hand expression .

  

Conclusion:

Hence, it is proved that and .

x.

To prove: The vector property using component form.

Given:

An expression .

Calculation:

Suppose that .

Substitute the assumed values in left hand expression .

  

Obtain the magnitude of $〈a{u}_1,a{u}_2〉$ .

  

Conclusion:

Hence, it is proved that .