Vector Projection Formula Derivation: Properties & Dot Product

Vector projection is defined when a vector is resolved into its two components, one is parallel to the second vector and the other is perpendicular to the second. A vector is a quantity whose magnitude and direction are represented by an arrow over its symbol. The vector does not change when we move a body parallel to itself. It does not have a definite location, despite its magnitude and direction. The orthogonal projection of a vector ‘a’ on another non-zero ‘b’ vector is the first vector's projection on a straight line parallel to the second vector. Projb ‘a’ denotes the projection of vector an onto the b formula.

Read More: Difference between scalar and vector

Key Terms: Vector projection,Vector a's scalar projection, dot product, angle between the two vectors, vectors, components, magnitude

---

Projection Vector

[Click Here for Sample Questions]

The length of a given vector's shadow cast over another vector is the vector projection of one vector over another vector. It is calculated by multiplying the magnitude of the two vectors by the cosecant of the angle between them. A scalar value is the result of a vector projection formula.

Projection vector

A vector projection a over b is obtained by the angle between the vectors a and b is determined by multiplying vector a by the Cosecant of the angle between the vectors a and b. To acquire the final value of the projection vector, this is simplified even more. The magnitude of the projection vector is equal to that of vector ‘a’, and its direction is the same as that of vector ‘b’.

Read More: Resolution of Vectors

---

Projection Vector Formula

[Click Here for Previous Year Questions]

When a vector is resolved into its two components, one is parallel to the second vector and the other is perpendicular to the second, thus, vector projection is defined. With the help of two vectors, let's call them a and b, we'll define the vector projection formula. Consider the diagram below:

Projection Vector Formula

There are two vectors, a and b, in the diagram above, and is the angle between them. Then the vector projection is as follows:

Proj_ba = \(\frac{\overrightarrow{a} . \overrightarrow{b}}{(b)^2} \overrightarrow{b}\)

The '.' operator defines the dot product of vectors a and b.

Vector a's scalar projection on b is given by:

a₁=||a||cosθ

The angle formed by a vector a and another vector b is shown below. a1 is the scalar factor.

In addition, vector projection is defined by

a₁ = a₁bˆ= (||a||cosθ)b

The formula for projecting a vector ‘a’ onto a vector ‘b’ produces a vector with the same direction as vector b.

Do check out:

---

Derivation of Projection Vector Formula

[Click Here for Sample Questions]

Assume that OA = a and OB = b, and the angle between a and b. It exists as a component of vector ‘a’ across vector ‘b’. 

Draw AL perpendicular to OB.

From the right-angled triangle OAL, cosθ = OL/OA

OL=OA cosθ

OL=|a| cosθ

OL is considered as the projection vector of vector ‘a’ on vector ‘b’.

\(\overrightarrow{a} \) . \(\overrightarrow{b} \) = |\(\overrightarrow{a} \)| . |\(\overrightarrow{b} \)| cosθ

\(\overrightarrow{a} \) . \(\overrightarrow{b} \) = |\(\overrightarrow{b} \)| ( |\(\overrightarrow{a} \)| . cosθ)

\(\overrightarrow{a} . \overrightarrow{b} \) = | \(\overrightarrow{b} \) | OL

OL = \(\frac{\overrightarrow{a} . \overrightarrow{b}}{|\overrightarrow{b}|}\)

Hence, the vector projection formula of vector ‘a’ on vector \('b' = \frac{\overrightarrow{a} . \overrightarrow{b}}{|\overrightarrow{b}|}\)

Like this, the vector projection of vector ‘b’ on vector \(\frac{\overrightarrow{a} . \overrightarrow{b}}{|\overrightarrow{a}|}\)

Read More: Vector product of two vectors

---

Properties of Vector Projection Formula

[Click Here for Sample Questions]

The projection properties are as follows,

• When the angle is 90 degrees, a1 equals 0.
• If 90° < θ ≤ 180° b and a1 are in the opposite direction.
• a1 and vector b have the same direction if 0 ≤ θ < 90°.
• In Gram–Schmidt orthonormalization, the concept of vector projection is often used. The notion can also be used to determine if two convex objects meet or not.

Read More: Class 12 Mathematics Chapter 10 Vector Algebra

---

Angle between Two Vectors

[Click Here for Previous Year Questions]

The cosine of the angle between two vectors is used to compute the angle between them. The dot product of the individual constituents of the two vectors divided by the product of the magnitude of the two vectors equals the cosine of the angle between them. The angle between the two vectors can be calculated using the following formula.

Cos θ  = \(\frac{\overrightarrow{a} . \overrightarrow{b}}{|\overrightarrow{a}| ||\overrightarrow{b}|}\)

Cos θ  = \(\frac{a1.b1 + a2.b2 + a3.b3}{\sqrt{(a1)^2 + (a2)^2 +(a3)^2}.\sqrt{(b1)^2 + (b2)^2 + (b3)^2}}\)

Read More: Properties of Vector Addition

---

Dot Product of Two Vectors

[Click Here for Sample Questions]

The scalar product of two vectors is sometimes known as the dot product. The scalar product is generated by expressing the two vectors in terms of unit vectors, i, j, k, along the x, y, and z axes:

Consider,

\(\overrightarrow{a} \) = \(a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\)

\(\overrightarrow{b}\) = \(b_1\hat{i} + b_2\hat{j} + b_3\hat{k}\)

\(\overrightarrow{a} . \overrightarrow{b} \) = (\(a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\)) (\(b_1\hat{i} + b_2\hat{j} + b_3\hat{k}\))

\(\overrightarrow{a} . \overrightarrow{b} \) = a₁b₁ + a₂b₂ + a₃b₃

---

Things to Remember

• Vector projection is defined when a vector is resolved into its two components, one is parallel to the second vector and the other is perpendicular to the second. 
• A vector is a quantity whose magnitude and direction are represented by an arrow over its symbol. The vector does not change when we move a body parallel to itself.
• A vector projection a over b is obtained by the angle between the vectors a and b is determined by multiplying vector a by the Cosecant of the angle between the vectors a and b. 
• The magnitude of the projection vector is equal to that of vector a, and its direction is the same as that of vector b.
• The dot product of the individual constituents of the two vectors divided by the product of the magnitude of the two vectors equals the cosine of the angle between them.

---

Previous Year's Questions

1. If aa is a unit vector, then |a×^i|2+|a×^j|2+|a×^k|2=|a×i^|2+|a×j^|2+|a×k^|2= …….[TS EAMCET 2017]
2. Let →a,→ba→,b→ & →cc→ be non-coplanar unit vectors equally inclined to one another at an acute angle  …..[BITSAT 2017]
3. Two vectors are given by →A=3^i+^j+3^kA→=3i^+j^+3k^ and →B=3^i+5^j−2^kB→=3i^+5j^−2k^.  ….[JKCET 2007]
4. A constant power is supplied to a rotating disc. The relationship between the angular velocity ω  of the disc and number of  ...[CBSE CLASS XII]
5. From a point A with position vector p(^i+^j+^k)p(i^+j^+k^), AB and AC are drawn perpendicular to the….[JEE Main 2018]
6. If a|(b+c)a|(b+c) and a|(b−c)a|(b−c) where a,b,c∈Na,b,c∈N then…...[KCET 2006]
7. The vector equation of the symmetrical form of equation of straight line x−53=y+47=z−62x−53=y+47=z−62 is…...[VITEEE 2018]
8. If a and b are vectors such that |a+b|=|a−b||a+b|=|a−b| then the angle between a and b is….[KCET 2007]
9. Let →a=^i+^j+√2^k,→b=b1^i+b2^j+√2^ka→=i^+j^+2k^,b→=b1i^+b2j^+2k^ and →c=5^i+^j+√2^kc→=5i^+j^+2k^ be…..[JEE Main 2019]
10. Let →a=^i−2^j+^ka→=i^−2j^+k^ and →b=^i−^j+^kb→=i^−j^+k^ be two vectors. If →cc→ is a...  [JEE Main 2020]
11. If →aa→ and →bb→ are non-collinear vectors, then the value of a for which the vectors …..[JEE Main 2013]
12. If a unit vector →aa→ makes angles π/3π/3 with ^i,π/4i^,π/4 with ^jj^ and θ∈(0,π)θ∈(0,π) with ^kk^, then a value of θθ is : – [JEE Main 2019]
13. Let u,vu,v and ww be vectors such that u+v+w=0.u+v+w=0. If |u|=3,|v|=4|u|=3,|v|=4 and |w|=5|w|=5 then u⋅v+v⋅w+w⋅uu⋅v+v⋅w+w⋅u is equal to….[KEAM]
14. If the scalar product of the vector ˆi+ˆj+2ˆk with the unit vector along mˆi+2ˆj+3ˆk is equal to 2, then one of the values of m is...[KEAM]
15. Let A(1,−1,2)A(1,−1,2) and B(2,3,−1)B(2,3,−1) be two points. If a point P ...[KEAM]
16. The vectors of magnitude a,2a,3aa,2a,3a meet at a point and their directions are along the  ...[KEAM]
17. If →a, →b, →ca→, b→, c→ are non-coplanar and (→a+λ→b).[(→b+3→c)×(→c×4→a)]=0,(a→+λb→).[(b→+3c→)×(c→×4a→)]=0, then  …[KEAM]
18. If λ(3^i+2^j−6^k)λ(3i^+2j^−6k^) is a unit vector, then the values of λλ are….[KEAM]
19. If the projection of the vector →aa→ on →bb→ is →aa→ on →bb→ is |→a×→b||a→×b→| and if 3→b=→i+→j+→k,3b→=i→+j→+k→, then the angle  …..[KEAM]