Sample Problems on Parallel Vector

Example 1: Verify whether the vectors are parallel , antiparallel or intersecting U=3i + 2j -k and 3V =i + 2j – k.

Solution:

For vector U, the direction vector [Tex]\vec{u}[/Tex] is <3, 2, -1>.

For vector V, the direction vector [Tex]\vec{v}[/Tex] is <1, 2, -1>.

Comparing the direction vectors, we see that [Tex]\vec{u}[/Tex] and [Tex]\vec{v}[/Tex] are not scalar multiples of each other, nor are they negatives of each other.

Hence, U and V are neither parallel nor antiparallel. They intersect at some angle.

Example 2: Find a Unit Vector parallel to given vector U = 3i + 4j + 12j?

Solution:

Unit vector parallel to given vector can be found by dividing the given vector with it own magnitude as:

Given U = 3i + 4j + 12j ,

|U| = √3² + 4² +12²

= √9 + 16 + 144 = √169 = 13

So Unit vector parallel to given U vector is = (3i + 4j + 12k ) / 13

Example 3: Find the dot and cross product of the vectors U = 3i + j -2k and U = 6i +2j – 4k?

Solution:

Given,

U = 3i + j -2k

V = 6i + 2j – 4k

It can be seen that V = 2U

So we can say that both the vectors are parallel to each other.

Dot product : For parallel vector U.V = |U| |V|

or |U|=√3² + 1² +2² = √9 + 1 + 4 = √14

|V| = √6² +2² + 4² = √36 + 4 + 16 = √56

So U.V = √14 ×√56= √784 = 28.

Cross Product : The cross of two parallel vectors is zero.

Example 4: Find a vector parallel to the vector U = 3i + 4j and has magnitude twice the magnitude of U?

Solution:

Given vector U = 3i + 4j.

Magnitude of U, |U| = √(3² + 4²) = 5.

A vector parallel to U with twice the magnitude: V = 2U = 6i + 8j.

Magnitude of V, |V| = √(6² + 8²) = 10.

Parallel vectors are considered one of the most important concepts in vector algebra. When two vectors have the same or opposite direction, they are said to be parallel to each other. Note that parallel vectors can differ in magnitude, and two parallel vectors can never intersect each other. They are widely used in mathematics, physics, and other areas of engineering for defining lines and planes, representing force and velocity, and analyzing various structures.

In this article, we will learn about parallel vectors, the dot product, and the cross product of parallel vectors, as well as their properties, in detail.