The magnitude of a displacement vector and the magnitude of a velocity vector each have their own name. What are they?

The magnitude of a displacement is a distance , and the magnitude of a velocity is a speed . In each case taking the magnitude keeps the size and drops the direction, which is what turns the vector into a scalar.

International A Level (IAL)MathsEdexcelMechanics 1FlashcardsMathematical Models & VectorsWorking with Vectors

Working with Vectors (Edexcel International A Level (IAL) Maths: Mechanics 1): Flashcards

Exam code: YMA01

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FrontWorking with Vectors
Define a unit vector, and say what $i$ and $j$ are.

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Define a unit vector, and say what $i$ and $j$ are.
A unit vector is a vector whose magnitude is 1.
$i$ and $j$ are the unit vectors in the positive horizontal and positive vertical directions, and they are perpendicular to each other.
The column vector $(\begin{matrix} 3 \\ - 7 \end{matrix})$ can also be written $3 i - 7 j$ . What does each of the two numbers tell you?
The 3 is the number of units moved in the positive horizontal direction, so 3 units to the right.
The $- 7$ is the number of units moved vertically, and the minus sign makes it 7 units down.
The two forms mean exactly the same thing, but a final answer should be given in $i$ , $j$ form, with column vectors kept for the working.
Fill in the blanks to complete the formula for the magnitude of a vector:
$|x i + y j| = \sqrt{______+______}$
The completed formula is:
$|x i + y j| = \sqrt{x^{2} + y^{2}}$
It is Pythagoras' theorem: the two components are the perpendicular sides of a right-angled triangle, and the vector itself is the hypotenuse.
Two forces are given in $i$ , $j$ form. How do you find their resultant, and what does the resultant represent?
Add the $i$ components together and add the $j$ components together, collecting like terms exactly as you would in algebra.
The resultant is the single force that has the same effect on the particle as the two original forces acting together.
A resultant force is $(- 3 i + 9 j)$ N. Find its magnitude, leaving your answer as a simplified surd.
Square both components, add them and take the square root:
$\sqrt{{(- 3)}^{2} + 9^{2}} = \sqrt{90} = 3 \sqrt{10} N$
The minus sign disappears when the component is squared, so a magnitude comes out positive whichever way the force acts.
True or False?
If two position vectors have the same $j$ component, the one with the greater $i$ component is due east of the other.
True.
Equal $j$ components mean the two points are the same distance north, so the only thing separating them is how far east each one is.
The greater $i$ component is therefore further east, and the direction from the other point to it is due east.
The magnitude of a displacement vector and the magnitude of a velocity vector each have their own name. What are they?
The magnitude of a displacement is a distance, and the magnitude of a velocity is a speed.
In each case taking the magnitude keeps the size and drops the direction, which is what turns the vector into a scalar.
You need the direction of a vector as a bearing. Which unit vector points north, which points east, and how is a bearing measured?
$j$ points north and $i$ points east.
A bearing is measured from north, turning clockwise.
So a particle moving due north has a velocity of the form $k j$ with $k > 0$ , and one moving due east has a velocity of the form $k i$ .
Why should you sketch a vector before using $\tan^{- 1}$ to find its direction?
Because $\tan^{- 1}$ of the two components only gives you the acute angle the vector makes with the horizontal.
The sketch shows which way the vector actually points, and so what has to be done to that acute angle.
For $- 3 i + 9 j$ the acute angle is $\tan^{- 1} (\frac{9}{3}) = 71.6 °$ , and the bearing turns out to be $270 + 71.6 = 341.6 °$ .

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