Test: Vectors Multiplication - ACT MCQ

√49 = 7. When î + 14ĵ is multiplied by 7, we get 7î + 98ĵ as the answer. Hence, 7î + 98ĵ is the correct answer.

Any vector a, can be written as a =  Hence, when we multiply it by a scalar, the magnitude multiplies by that much amount.

Unit vector along X is î and that along Y is ĵ. Hence, eleven times unit vector along X added to 7 times unit vector along Y will give 11î + 7ĵ.

When 2î + 7ĵ is multiplied by 5, both the components get multiplied by 5. Hence, the answer is 10î + 35ĵ.

Unit vector along X is î and that along Y is ĵ. Hence, when the car travels 5m in X direction and 7m in y direction, the final position becomes 5î + 7ĵ.

A unit vector has the magnitude 1. It is used to represent non-unit vectors using along other unit vectors. Any vector can be represented as a sum of unit vectors, multiplied by required scalars.

When 5î + 10ĵ is divided by 5, both the components get divided by 5. Hence, the answer is î + 2ĵ.

Concept:

• The dot product is the sum of the products of the corresponding entries of the two sequences of numbers.
• Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them.

The Dot product of two vectors is defined as:

The Cross/Vector product of two vectors is defined as:

Explanation:

The dot product of two vectors is given by:

So option 2 is correct.

Some important results:

î × ĵ = k̂ and ĵ × î = -k̂

k̂ × î = ĵ and î × k̂ = -ĵ

ĵ × k̂ = î and k̂ × ĵ = -î

î × î = ĵ × ĵ = k̂ × k̂ = 0

î.ĵ = ĵ.k̂ = î.k̂ = 0

î.î = ĵ.ĵ = k̂.k̂ = 

Concept:

• Vector Product (cross product): Its magnitude is equal to the products of the magnitude of two vectors and sine of the angle between them and whose direction is perpendicular to the plane of the two vectors 

Mathematically it is written as: 

Where θ is the angle between a vector 

• Torque, Angular momentum, and Instantaneous velocity all can be expressed as the vector products of two vectors.

Explanation:

• Torque (τ): The torque acting on a particle is equal to the vector products of its position vector  and force vector 
  
• Angular momentum (L): It is equal to the cross product of its position vector  and the linear momentum 
  
• Linear Velocity (v): It is equal to the cross product of its angular velocity ω and the position vector 
  
• These physical quantities can be expressed as the vector products of two vectors. 
• Hence all of the above options are correct.

Concept:

• Vector quantities are those physical quantities that have both magnitude and direction and obey the vector law of addition.
  • Example Displacement, Velocity, Force, Momentum, etc

The magnitude of resultant of two vectors is given by:

Where A and B are the magnitudes of two vectors and θ is the angle between them
Calculation:

Let's assume two forces A and B having two equal force have their resultant equal to either.

A = B = R

cosθ = -1/2,  Since (cos -1/2 = 120° )

The angle between them is, θ = 120°