My teacher said that $\vec{A} = \left| \vec{A} \right| \hat{A}$ , where $\left| \vec{A} \right|$ is the magnitude and  is the direction of the vector.
In this homework question, what exactly do you mean by $A$? Is $A$ just another way to write $\left| \vec{A} \right|$ or is it something else?
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2$\begingroup$ This notation usually does refer to the magnitude of a vector and in this case $A$ does refer to the magnitude of $\vec A$ and similar for $B$ and $C$. $\endgroup$– joseph hCommented Mar 14, 2022 at 7:14
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$\begingroup$ @josephh so A, B and C have to be positive right? They can't just be any integer right? $\endgroup$– Pumpkin_StarCommented Mar 14, 2022 at 7:26
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$\begingroup$ @josephh But my teacher said that by convention, magnitudes are always positive $\endgroup$– Pumpkin_StarCommented Mar 14, 2022 at 8:03
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$\begingroup$ What I mean is B and C are not equal to zero. It is a trivial case, but vectors can be zero, though I’m guessing your teacher means they are never zero. Just ignore my last comment. You are correct. $\endgroup$– joseph hCommented Mar 14, 2022 at 8:07
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3$\begingroup$ Just so you know, I am be very annoyed by the inconsistency in how this question is written. It starts off distinguishing clearly between $\vec{A}$ and $|\vec{A}|$, then introduces a third notation $A$ without mentioning what it means. We can make guesses about what it means, but it's bad form on the question writer to leave us guessing like this. You should not feel bad about not knowing what it means right away. $\endgroup$– Luke PritchettCommented Mar 14, 2022 at 12:25
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1 Answer
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For a vector $\vec{R} = R_x \hat{\imath} \ + R_y \hat{\jmath} \ + R_z \hat{k}$, the magnitude is denoted by either $\left|\vec{R}\right|$ or simply $R$.
Because the magnitude is defined as $$\sqrt{{R_x}^2 + {R_y}^2 + {R_z}^2}$$ and as $\sqrt{A^2}$ = $|A| \geq 0$, the magnitude of a vector is necessarily non-negative.
If a vector is represented by a linear combination of two other vectors such as $\vec{C} = \vec{A} + \vec{B}$, then the magnitude of $\vec{C}$ is given by $$C = \sqrt{A^2 + B^2 + 2(\vec{A} \cdot \vec{B})}$$
where $\vec{A} \cdot \vec{B}$ represents the scalar product of the two vectors
Hope this helps.
