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Wize University Physics Textbook (Master) > Force Vectors (Advanced Info)

Vectors and Their Components

Forces Cartesian Vector Form (3D)Previous Section
Similar Triangles / “That Small Triangle” / Pythagorean TriplesNext Section
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Vectors and Their Components


  • A Vector is a quantity that has both magnitude and direction such as velocity.
  • A Scalar has magnitude but no direction such as speed.
  • Two vectors are equal to each other as long as they have the same size and direction
  • Vectors can be written in terms of their components based on different coordinate systems
  • The most common coordinate system is XYZ -coordinate system which is also known as Cartesian coordinate system.
  • Theses components are the shadow of the vector on different axis of the coordinate system
  • A given vector a⃗\vec{a} in 2D space is shown as:
a⃗=(ax,ay)=axi^+ayj^\boxed{\vec{a}=\left( a_x,a_y\right )=a_x\hat{i}+a_y\hat{j} }

- where vectors , i^\hat{i} and j^\hat{j} are basic unit vectors in xx and yy directions, respectively. These unit vectors are:
i^=(1,0) j^=(0,1)\begin{array}{l}\hat{i}=(1,0)\\\ \hat{j}=(0,1)\end{array}
  • Unit vectors are vectors with length equal to 1 which are used to show directions.
  • axa_xand aya_yare called components of vector a⃗\vec{a}




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ٰVector's Magnitude

Magnitude of a 2D Vector

We can calculate the magnitude (a.k.a. length or size) of a vector in 2D using Pythagorean's Theorem:

Wize Tip
The length of the vector v⃗=[v1, v2]\vec{v}=\left[v_1,\ v_2\right] is a scaler and is equal to:
∣v⃗∣=∣[v1, v2]∣=v1 2+v2 2|\vec v|=|[v_1,\ v_2]|=\sqrt{v_1^{\ 2}+v_2^{\ 2}}


Magnitude of a 3D Vector

Similarly, We can calculate the magnitude of a vector in 3D using an extension of this formula.

Wize Tip
The length of the vector u⃗=[u1, u2, u3]\vec{u}=\left[u_1,\ u_2,\ u_3\right] is
∣u⃗∣=∣[u1, u2, u3]∣=u1 2+u2 2+u3 2|\vec{u}|=|[u_1,\ u_2,\ u_3]|=\sqrt{u_1^{\ 2}+u_2^{\ 2}+u_3^{\ 2}}





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Components of a Vector Using the Angle It Makes Respect to x or y Axis


In many problems we need to find the components of a vector along different directions. For these problems we use trigonometric relationship using right triangle formed by the vector as the hypotenuse and components as the two other sides of the triangle.



Example:
Consider the following vector which makes an angle of θ\thetarespect to the x-axis (horizontal direction)


Components of this vector along x and y directions could be found as:
vx=∣v⃗∣cos⁡θvy=∣v⃗∣sin⁡θ\begin{array}{l}v_x=|\vec{v}|\cos\theta\\v_y=|\vec{v}|\sin\theta\end{array}

Watch Out!
  • Do not always assume that "x is cosine, y is sine". This depends completely on where we define the angle θ\theta. Normally, θ\thetais defined as an angle relative to the horizontal (x) axis, and these conventions will work, but when the angle is relative to the vertical (y) axis, the convention is flipped.



We can find the length of this vector and the angle it makes respect to horizontal x-axis using the following equation:
  • Length of a vector in 2-D space is equal to:
∣v⃗∣=vx2+vy2\boxed{|\vec{v}|=\sqrt{v_x^2+v_y^2}}
  • The direction of a vector usually is reported by the angle it makes with positive xx-axis (Counter-clockwise) using following formula:
θ=tan⁡−1(vyvx)\boxed{\theta=\tan^{-1}\left(\frac{v_y}{v_x}\right)}



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Example: Vector Connecting Two Points


If A(1,2)A\left(1,2\right) and B(−1,3)B\left(-1,3\right), calculate AB→\overrightarrow{AB}, length of AB→\overrightarrow{AB} and the angle it makes with positive xx-axis.

Solution:

AB→=B→−A→ =(−1,3)−(1,2)=(−1−1,3−2)=(−2,1)\begin{array}{rl}\overrightarrow{AB}=\overrightarrow{B}-\overrightarrow{A}\ = &\left(-1,3\right)-\left(1,2\right)\\=&(-1-1,3-2)=(-2,1)\end{array}




∣AB→∣=(−2)2+12=5\left|\overrightarrow{AB}\right|=\sqrt{\left(-2\right)^2+1^2}=\sqrt{5}

 θ=180−tan⁡−1∣AByABx∣=180−tan⁡−1∣1−2∣=153.43°\ \theta=180-\tan^{-1}\left|\frac{AB_y}{AB_x}\right|=180-\tan^{-1}\left|\frac{1}{-2}\right|=153.43\degree


Exam Tip
*With these types of examples, instead of memorizing some general rule, it's best to draw the picture and reason about whether to add 180, subtract from 180, add 90, etc, depending on what you drew.

A football is thrown with a horizontal velocity of 10m/s and a vertical velocity of 4.25 m/s. What is football's velocity?