Vector Algebra

Vectors
, and are coplanar 







So,
Thus = 0 for all values of x

The term "vectors" or "vector quantities" refers to quantities that have both magnitude and direction. The items known as vectors are those that may be discovered in cumulative form in vector spaces using two different types of operations. These operations in the vector space include multiplying the vector by a scalar quantity and adding two vectors together. The result still exists in the vector space even if these operations might change the vector's ratios and order. It is frequently identified by symbols like U, V, and W.

A directed line is one that has an arrowhead. The size and direction of a section of the directed line are both present. A vector is the name for this section of the directed line. It is shown as a or, more frequently, as AB. A and B are the line's beginning and ending points, respectively, in this segment of the line AB.

Since ,

....(i) 
Now,

(Using (i)) 

A vector is a unit of measure that lacks a location but has both magnitude and direction. Examples include speed and acceleration. A vector is a directed line segment with an arrow indicating its direction and a length equal to the vector's magnitude. The vector is often represented by an arrow whose length is proportional to the magnitude of the quantity and whose direction is comparable to that of the quantity. Additionally, scaled vector diagrams are typically used to depict vector quantities. The displacement vector is represented in the vector diagram.

As was already established, the vector is essentially represented by an arrow, whose length is directly proportional to the magnitude of the quantity and whose direction is the same as the quantity. The displacement vector will be shown in the vector diagram:

Vectors are the name given to the mathematical representation of physical quantities that include both the magnitude and the direction.

Vectors must have their resultant independently determined since they are added geometrically rather than algebraically.

This vector's magnitude is expressed as either |ab| or |a|. With the aid of the Pythagorean theorem, it illustrates the vector length and shows how every physical quantity's vector may be represented as a straight line with an arrowhead. The length of the straight line indicates the vector's magnitude, and the arrowhead indicates its direction. 

The diagonal of a parallelogram formed from the same point will describe the outcome if the two forces are represented as vectors a and b by the neighboring sides of the parallelogram. Vector (r = a + b) is the rule of addition for parallelograms.

Three vectors are combined to form the scalar triple product of vectors. Because it evaluates to a single integer, much like the dot product, it is a scalar product. It requires multiplying the dot product of one of the vectors by the cross product of the other two vectors. It is expressed mathematically as (a x b)xc. The scalar triple product can be used to indicate the volume of a parallelepiped. The dot product of one vector and the cross product of the other two vectors is the scalar triple product, sometimes referred to as the mixed product, box product, or triple scalar product.

When vectors are multiplied and a scalar triple product is formed, the formula for the triple product of three vectors, or the dot product of a vector with the cross product of the other two vectors, is known as the scalar triple product formula. It is written as follows:

[a b c] = (a x b).c

In this instance (a x b). c is a scalar quantity and the resultant vector. This formula can alternatively be expressed by swapping the cross and dot in the middle, as seen below (a x b). c is the same as a. (b x c). The scalar triple product of vectors is obvious from its name: it is the product of three vectors. It requires multiplying one of the vectors by the cross product of the other two.

The properties of scalar triple product of vectors are:

• If the vectors are permuted cyclically, then (a × b) . c = a.( b × c)
• If the vectors are permuted cyclically, then a.(b × c) = b.(c × a) = c.(a × b)
• If the triple product of vectors is zero, it can be assumed that the vectors are coplanar.

To determine the value of that makes vectors , , and coplanar:

Given Vectors:

1. Coplanarity Condition:
Three vectors are coplanar when their scalar triple product equals zero:

2. Compute Cross Product :

3. Calculate Dot Product:

4. Solve for :
Set the scalar triple product to zero:

Final Answer:
The value of that makes the vectors coplanar is: