Chapter 1: Discussion Questions (Q1.16 - Q1.20)

NOTE: I learned how to implement LaTeX in this blog, so I will be using it from here on out!

Q1.16 Can you find a vector quantity that has a magnitude of zero but components that are different from zero? Explain. Can the magnitude of a vector be less than the magnitude of any of its components?

Response: The magnitude of a vector is given by $A = \sqrt{(A_x)^2+(A_y)^2+(A_z)^2}$ Since each term under the radical is squared, each term under the radical must be positive, or zero. Therefore, the radicand is zero only if all the components are zero. So a vector quantity with a magnitude of zero must have all components equal to zero as well. The magnitude must be greater than or equal to any single component. If all three components are zero, then the magnitude is zero and equal to all 3 components. If just one component is nonzero, ($A_x$, for example) then $A = \sqrt{(A_x)^2 + 0 + 0} = A_x$. A similar argument can be made for $A_y$ and $A_z$. If two components are non-zero, then $A$ must be greater than any single component, since a positive quantity is being added to the radicand.

Q1.17 (a) Does it make sense to say that a vector is negative? Why? (b) Does it make sense to say that one vector is the negative of another? Why? Does your answer here contradict what you said in part (a)?

Response: (a) It does not make sense to say that a vector is negative. That would imply a negative magnitude, but by definition a magnitude is always positive. (b) It does make sense to say that one vector is the negative of the other. The way to think of this is in terms of the additive inverse. In normal arithmetic, a number $x$ has an additive inverse, $-x$ (negative $x$) such that $x + -x = 0$. In the same vein, a vector $\bf \overrightarrow{A}$ has a vector additive inverse $\bf \overrightarrow{-A}$ such that $\bf \overrightarrow{A} + \bf \overrightarrow{-A} =$ 0 (A vector quantity with magnitude zero).


Q1.18 If $\bf \overrightarrow{C}$ is the vector sum of $\bf \overrightarrow{A}$ and $\bf \overrightarrow{B}$, $\bf \overrightarrow{C} = \bf \overrightarrow{A} + \bf \overrightarrow{B}$, what must be true if C = A + B? What must be true if C = 0?

Response:  If C = A + B, then $\bf \overrightarrow{A}$ and $\bf \overrightarrow{B}$ must be parallel vectors (same direction). If C = 0, then $\bf \overrightarrow{A}$ and $\bf \overrightarrow{B}$ must be anti-parallel (opposite directions) and have the same magnitude.


Q1.19 If $\bf \overrightarrow{A}$ and $\bf \overrightarrow{B}$ are nonzero vectors, is it possible for $\bf \overrightarrow{A}\cdot\bf \overrightarrow{B}$ and $\bf \overrightarrow{A}\times\bf \overrightarrow{B}$ both to be zero? Explain.

Response: No. The scalar product is zero when the two vectors are perpendicular. The vector product is zero when the two vectors are parallel or anti-parallel. Two vectors cannot be both perpendicular AND parallel (or anti-parallel) in euclidean space.


Q1.20 What does $\bf \overrightarrow{A}\cdot\bf \overrightarrow{A}$, the scalar product of a vector with itself, give? What about $\bf \overrightarrow{A}\times\bf \overrightarrow{A}$, the vector product of a vector with itself?

Response: The dot product is defined as $\bf \overrightarrow{A}\cdot\bf \overrightarrow{B} = A_xB_x + A_yB_y + A_zB_z$. If $\bf \overrightarrow{B} = \bf \overrightarrow{A}$, then the dot prodct $$\bf \overrightarrow{A}\cdot\bf \overrightarrow{A} = A_xA_x + A_yA_y + A_zA_z = (A_x)^2 + (A_y)^2 +(A_z)^2.$$ The cross product $\bf \overrightarrow{C} = \bf \overrightarrow{A}\times\bf \overrightarrow{B}$ gives $C_x = A_yB_z - A_zB_y$, $C_y = A_zB_x - A_xB_z$, and $C_z = A_xB_y - A_yB_x$. If $\bf \overrightarrow{B} = \bf \overrightarrow{A}$, then the cross product $\bf \overrightarrow{C} = \bf \overrightarrow{A}\times\bf \overrightarrow{A}$ gives the following components of $\bf \overrightarrow{C}$: $$C_x = A_yA_z - A_zA_y = 0,$$ $$C_y = A_zA_x - A_xA_z = 0,$$ $$C_z = A_xA_y - A_yA_x = 0.$$ That is $\bf \overrightarrow{C} = 0$.