Q1.4 Adding vectors (K)

Consider two vectors $\vec{A}$ and $\vec{B}$. Vector $\vec{A}$ points northeast; vector $\vec{B}$ points northwest. Each vector has magnitude $\sqrt{2}$. Let $\vec{C}= \vec{A} + \vec{B}$.
  1. What is the direction of $\vec{C}$? Draw a vector diagram showing the relationship between the three vectors. Deduce the magnitude of the vector $\vec{C}$.
  2. On your vector diagram add rectangular x and y coordinate axes representing east and north directions respectively. Mark unit vectors î and $\hat{j}$ on these axes. Read off the values of the components Ax ,Ay ,Bx , By. Deduce the magnitude and direction of $\vec{C}$ this way.
  3. What is C ? What is A+B ? Comment.
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Hint

It is often possible to determine the direction of a vector by inspection. In this case you should immediately be able to ‘see’ what the direction of $\vec{C}$ is.

The point of the last part is to get you to distinguish between adding vectors and adding the magnitudes of those vectors.

 

Solution

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Solution

  1. Once you have drawn $\vec{A}$ and $\vec{B}$, or even pictured them in your mind’s eye, you see immediately that $\vec{C}$ must be vertically upwards (or north).

    It is obvious isnt it? A physicist might well put a gloss on this by citing the ‘symmetry’ of the two vectors $\vec{A}$ and $\vec{B}$.

    Once you’ve decided this it is easy to see how the vector triangle fits together; to see that it is a right-angled triangle; and to deduce (Pythagoras) that

    \[ C= \sqrt{A^2 +B^2} =2 \]

    Graphic - No title - addvecs
  2. Constructing the axes and unit vectors as asked we identify:
    \[ \vec{A}=A_x \hat{i} +A_y\hat{j} \hspace{0.5cm}\mbox{\rm with}\hspace{0.5cm} A_x= 1 \hspace*{1cm} A_y =1 \]
    and
    \[ \vec{B}=B_x \hat{i} +B_y\hat{j} \hspace{0.5cm}\mbox{\rm with}\hspace{0.5cm} B_x= -1 \hspace*{1cm} B_y =1 \]
    It follows that
    \[ \vec{C}=C_x\hat{i} +C_y\hat{j} =(A_x+B_x)\hat{i} +(A_y+B_y)\hat{j} =2\hat{j} \]
    which expresses both the magnitude and the direction of $\vec{C}$.
  3. Finally while

    \[ C = \mid\vec{C}\mid =\mid\vec{A} +\vec{B}\mid = 2 \]
     
    \[ A + B = \mid \vec{A} \mid + \mid \vec{B} \mid = 2\sqrt{2} \]

    The point here is to emphasize the difference between vector and scalar addition. In my experience many students take a long time to learn this. If you are uncertain save yourself some pain by sorting it out now. Your tutor is there to help!