Vector projection is a fundamental concept in physics and mathematics that describes how one vector influences another along a specific direction. It can be visualised as the shadow that one vector casts onto another when light is shone perpendicular to the second vector.
When working with two vectors,
Vector projection actually comes in two forms:
1. Scalar Projection
This is the length of the projection — a number that tells us how much of
\text{Scalar projection of } \ \overrightarrow{a} \text{ on } \overrightarrow{b} = |\overrightarrow{a}| \cos \theta
where θ is the angle between vectors a and b.
2. Vector Projection
This is the actual vector that represents the projection of
\text{Vector projection of } \ \overrightarrow{a} \text{ on } \overrightarrow{b} = (\frac{\overrightarrow{a} \cdot \overrightarrow{b}}{|\overrightarrow{b}|^2 })\overrightarrow{b}
Example of Vector Projection
Pushing a Box on the Floor: Imagine you’re pushing a box at an angle to the ground using a force F.
Not all of your applied force moves the box forward; only the horizontal component of the force (the projection of F on the horizontal direction) contributes to motion. The vertical part just presses the box into the floor. This horizontal projection is Fcosθ
Vector Projection Formula
If
Projection of Vector a on Vector b
Proj\ _ba= \frac {a.b}{|b|} a.b = Dot product of
\vec a and\vec b |b| = magnitude of
\vec b
Derivation of Vector Projection Formula
Let us assume, OP =
\vec A and OQ =\vec B and the angle between OP and OQ is θ. Drawn PN perpendicular to OQ.In the right triangle OPN, Cos θ = ON/OP
⇒ ON = OP Cos θ
⇒ ON = |\vec A | Cos θON is the projection vector of
\vec A on\vec B
\vec A.\vec B = |\vec A||\vec B|cos \theta
⇒\vec A.\vec B = |\vec B(|\vec A||cos \theta)
⇒\vec A.\vec B = |\vec B|ON
⇒ ON =\frac{\vec A.\vec B}{|\vec B|}
Hence, the ON =|\vec A|.\hat B Thus the Vector Projection of
\vec A on\vec B is given as\frac{\vec A.\vec B}{|\vec B|} The Vector Projection of
\vec B on\vec A is given as\frac{\vec A.\vec B}{|\vec A|}
Vector Projection Important Terms
To find the vector projection we need to learn to find the angle between two vectors and also to calculate the dot product between two vectors.
Angle Between Two Vectors
The angle between the two vectors is given as the inverse of the cosine of the dot product of two vectors divided by the product of the magnitude of two vectors.
Let's say we have two vectors
\vec A and\vec B angle between them is θ⇒ cos θ =
\frac{\vec A.\vec B}{|A|.|B|}
⇒ θ = cos-1\frac{\vec A.\vec B}{|A|.|B|}
Dot Product of Two Vectors
Let's say we have two vectors
\vec A.\vec B = (a_1\hat i + a_2\hat j + a_3\hat k)(b_1\hat i + b_2\hat j + b_3\hat k)
⇒\vec A.\vec B = a1b1 + a2b2 +a3b3
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Applications and Significance of Vector Projection
- Work and Force Analysis
Vector projection helps in finding the component of a force that acts in a particular direction. For example, when pushing an object at an angle, only the projected part of the force contributes to its movement along the surface. - Computer Graphics and Visualization
In 3D graphics, projection is used to simulate lighting, shadows, and reflections. By projecting light vectors onto surfaces, realistic rendering effects are achieved. - Game and Animation Design
Motion simulation in games and animations relies on projecting movement vectors onto terrain surfaces, ensuring natural and realistic character movement. - Navigation and Path Optimization
GPS systems use vector projection to compute the shortest and most accurate path between two locations by projecting displacement vectors onto the Earth’s surface.
Vector Projection in Real-World
- GPS Navigation
Used to project displacement vectors between locations onto the Earth’s surface to determine optimal travel routes. - Sports Analytics
Helps in analyzing player and ball movements by projecting motion vectors onto the field or court to study speed, direction, and performance. - Renewable Energy (Wind Turbines)
Engineers project wind velocity vectors onto turbine blade planes to find the most effective orientation for maximum energy generation. - Augmented Reality (AR) Applications
Vector projection is used to align virtual elements accurately with physical environments, making digital overlays appear realistic.
Vector Projection Formula Examples
Example 1. Find the projection of a vector
Solution:
Here,
\vec{a}=4\hat i + 2\hat j + \hat k \\\vec{b}=5\hat i -3\hat j + 3\hat k .We know, projection of Vector a on Vector b =
\frac{\vec{a}.\vec{b}}{|b|}
\dfrac{(4.(5) + 2(-3) + 1.(3))}{|\sqrt{5^2 + (-3)^2 + 3^2}|}\\=\dfrac{17}{\sqrt{43}}
Example 2. Find the projection of the vector
Solution:
Here,
\vec{a}=5\hat i + 4\hat j + \hat k \\\vec{b}=3\hat i + 5\hat j - 2\hat k. We know, projection of Vector a on Vector b =
\frac{\vec{a}.\vec{b}}{|b|}
\dfrac{(5.(3) + 4(5) + 1.(-2))}{|\sqrt{3^2 + 5^2 + (-2)^2}|}\\=\dfrac{33}{\sqrt{38}}
Example 3. Find the projection of the vector
Solution:
Here,
\vec{a}=5\hat i - 4\hat j + \hat k \\\vec{b}=3\hat i - 2\hat j + 4\hat k. We know, projection of Vector a on Vector b =
\frac{\vec{a}.\vec{b}}{|b|}
\dfrac{(5.(3) + ((-4).(-2)) + 1.(4))}{|\sqrt{3^2 + (-2)^2 + (4)^2}|}\\=\dfrac{27}{\sqrt{29}}
Example 4. Find the projection of the vector
Solution:
Here,
\vec{a}=2\hat i - 6\hat j + \hat k \\\vec{b}=8\hat i - 2\hat j + 4\hat k We know, projection of Vector a on Vector b =
\frac{\vec{a}.\vec{b}}{|b|}
\dfrac{(2.(8) + ((-6).(-2)) + 1.(4))}{|\sqrt{8^2 + (-2)^2 + (4)^2}|}\\=\dfrac{32}{\sqrt{84}}
Example 5. Find the projection of the vector
Solution:
Here,
\vec{a}=2\hat i - \hat j + 5\hat k \\\vec{b}=4\hat i - \hat j + \hat k. We know, projection of Vector a on Vector b =
\frac{\vec{a}.\vec{b}}{|b|}
\dfrac{(2.(4) + ((-1).(-1)) + 5.(1))}{|\sqrt{4^2 + (-1)^2 + (1)^2}|}\\=\dfrac{14}{\sqrt{18}}
Example 6: Find the angle between the vectors
Solution :
\vec{a}=4\hat i + 3\hat j - \hat k \\\vec{b}=2\hat i - \hat j + 2\hat k cos θ =
\frac{\vec A.\vec B}{|A|.|B|} a.b = (4 x 2) + (3 x (-1)) + (( -1) x 2)
= 8 - 3 - 2 = 3|a| = √ (4) 2 + (3) 2 + (- 1) 2
|a| = √ 16 + 9 + 1 = √ 26|b| = √ (2) 2 + (- 1) 2 + (2) 2
|b| = √ 4 + 1 + 4 =|b|
= √ 9 = 3
Now computig value cos θcos θ = a.b/|a||b|
= 3 /3√26
=1/√26θ = cos-1 1/√26
θ =78.7
Unsolved Examples on Vector Projection
Question 1: Find the projection of vector
Question 2: If the project of
Question 3: The projection of vector
Find the ration of magnitudes
Question 4: Two vectors are given as
If the projection of
Question 5: Find the projection of vector
Example 6: Let
Find the projection vector (not just scalar) of