Solved Questions on Co-Initial Vectors

Question 1: Three vectors A, B, and C have magnitudes 5, 7, and 3 respectively. They are co-initial and their directions are inclined at angles of 30, 45 , and 60 respectively with the positive direction of the x-axis. Find the magnitude and direction of their resultant vector.

A= (5 cos30)i +(5sin30)j

B=(7 cos45)i+(7 sin45)j

C=(3 cos60)i + (3 sin60)j

now, we add up the x-component and y-component separately,

sum of x-components = (5 cos30) + (7 cos 45) + (3 cos60) = ( 5. √3/2) + (7. √2/2) + (3. 1/2)

sum of x-components = (5√3 + 7√2 + 3)/2

sum of y-components = (5 sin30) + (7 sin45) + (3 sin60) = (5. 1/2)+(7 . √2/2)+(3.√3/2)

sum of y-components = (5 + 7√2 + 3√3) / 2

Now, the magnitude of the resultant vector R is given by:

R=√(sum of x-components)² + (sum of y-components)²

=√((5√3 + 7√2 + 3)/2)² + ((5 + 7√2 + 3√3)/2)² = √ 137/4 + 7√3/2 + 21√2/4 + 5√6/2

Now , let’s find the direction of the resultant vector R,

tanθ= sum of y-components/sum of x-components = (5 + 7√2 + 3√3)/(5√3 + 7√2+3).

Question 2: Two vectors P and Q are co-initial. The magnitude of P is 8 units and the magnitude of Q is 5 units. If the angle between them is 60 , find the magnitude and direction of their resultant vector.

Given that P and Q are co-initial vectors with magnitudes ∣ P∣=8 units and ∣ Q∣=5 units, and the angle between them is 60 .

To find the resultant vector, we can use the law of cosines. The magnitude of the resultant vector can be calculated as:∣R∣²=∣P∣²+∣Q∣² −2∣P∣∣Q∣ cos(θ)

Where:

• ∣P∣ and ∣Q∣ are the magnitudes of vectors P and Q respectively.
• θ is the angle between vectors P and Q.

Substituting the given values, we get:

∣R∣² =(8)²+(5)² – 2(8)(5)cos(60)

∣R∣²=64+25−80×1/2

∣R∣²=64+25−40

∣R∣² =89−40

∣R∣² =49

∣R∣= √49

∣R∣=7.

So, the magnitude of the resultant vector is ∣R∣=7 units.

Now, to find the direction of the resultant vector, we can use trigonometry. The direction θ_R of the resultant vector with respect to vector P can be calculated using the law of sines:

sin(θ_r)= |Q|sin(θ)/|R|

Substituting the given values, we get:

sin(θ_r)=5sin(60)/7

sin(θ_r)=(5×√3/2)/7

sin(θ_r)=5√3/14

θr=sin^-1(5√3/14)

θ_r ≈ 53.13

So, the direction of the resultant vector with respect to vector P is approximately 53.13 .

Therefore, the magnitude of the resultant vector is 7 units and its direction with respect to vector P is approximately 53.13.

“Co-initial vectors” typically refers to vectors that share the same initial point or starting point in a vector space. In other words, if you have multiple vectors and they all begin at the same point or have the same starting position, they are said to be co-initial. In mathematics, vectors are often represented graphically as arrows, with the starting point being the head of the arrow and the ending point being the tail of the arrow.

When vectors share the same starting point, they can be conveniently analyzed together, especially when studying geometric properties or performing vector operations like addition or subtraction. Understanding co-initial vectors is crucial in various fields such as physics, engineering, and computer graphics, where vectors are used to represent forces, velocities, displacements, and other quantities. In this article, we’ll study what co-initial vectors are, their various properties, etc.