Question

Consider three points $P=(-\sin (\beta -\alpha ),-\cos \beta ),Q=(\cos (\beta -\alpha ),\sin \beta )$ and $R=(\cos (\beta -\alpha +\theta ),\sin (\beta -\theta ))$, where $0<\alpha ,\beta ,\theta <\frac{\pi }{4}.$ Then, (A) $P$ lies on the line segment $RQ$ (B) $Q$ lies on the line segment $PR$ (C) $R$ lies on the line segment $QP$ (D) $P,Q,R$ are non-collinear

Short answer

After comparing the coordinates, it is evident that none of the points P, Q, or R can be expressed as a linear combination of the other two, which implies that P, Q, and R are non-collinear.

Step-by-step solution

Analyze the Position of Point P

To verify if point P lies on the line segment RQ, one must check if P can be expressed as a linear combination of points R and Q where the coefficients add up to 1 and are non-negative. The coefficients represent how far P is from R and Q along the line segment.

Find a Linear Combination for P

Assume if such coefficients exist, it is possible to write P as $P=\text{t}(tR+(1-t)Q)$, for some $t$ in the interval $0\le t\le 1$. Compare both x and y coordinates of point P with the corresponding coordinates obtained by the linear combination and solve for $t$.

Substitute for Points R and Q

Substitute the x-coordinate and y-coordinate of P with $-\sin (\beta -\alpha )$ and $-\cos (\beta )$, respectively. For R and Q, use the given points in the equation of Step 2 and equate both coordinates to form two equations with $t$ as the unknown.

Analyze the system of equations for t

Check if the obtained system of equations has a solution for $t$ in the interval $0\le t\le 1$. If it does, then P lies on the line segment RQ.

Repeat for Point Q and R

Using the same methodology, examine whether Q lies on the line segment PR or R lies on the line segment QP by checking if they can be expressed as a linear combination of the other two points.

Determine the Position of P, Q, and R

If none of the points P, Q, or R can be written as a linear combination of the other two points within the constraints of the coefficients, then the points P, Q, and R are non-collinear.

Key concepts

Linear Combination

A linear combination in mathematics is a concept where two or more vectors (in this context, points) are multiplied by scalar coefficients and added together to create a new vector. It is represented by an equation like $v=a_1{v}_1+a_2{v}_2+\cdots +a_n{v}_n$ where $a_i$ are scalars, and $v_i$ are vectors.

In the context of coordinate geometry and our exercise, the points P, Q, and R are taken as vectors in 2D space. To determine if a point lies on the line segment joining two other points, we need to see if it can be expressed as a linear combination of those two points with particular conditions on the coefficients. These conditions—namely that the coefficients are non-negative and sum up to one—ensure that the resulting point indeed lies between (or is one of) the original points.

For our specific problem, if point P were on the line segment RQ, there would exist a $t$ such that P can be written as $P=tR+(1-t)Q$, with $0\le t\le 1$. The coefficients $t$ and $1-t$ tell us how 'far' P is from Q and R, respectively.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where algebraic equations are used to study geometric shapes and analyze positional relationships between figures in a coordinate plane. The basic elements of this mathematical field are points, which have coordinates that define their position, and lines, which can be defined by linear equations or expressions involving points.

In problems like our exercise, we use principles of coordinate geometry to deduce relationships between various points based on their coordinates. For example, to check if three points are collinear (lie on a single straight line), we examine the coordinates and derive certain equalities that must hold. The concept of slopes and linear combinations derived from the coordinates of points also plays a pivotal role in deducing such geometric relationships.

Trigonometry in Coordinate Geometry

Trigonometry in coordinate geometry involves using trigonometric ratios and identities to solve problems involving geometric figures in a plane. By incorporating trigonometry, we can deal with angles and distances, which are fundamental in analyzing the properties of triangles and other shapes defined by points on the coordinate plane.

In our specific problem, the coordinates of points P, Q, and R involve trigonometric functions of angles $\alpha ,\beta ,\text{and}\theta$. This presents an interesting challenge because we need to utilize trigonometric identities to simplify expressions and verify the positions of these points. For instance, we can use trigonometric identities to transform the coordinates of P, Q, and R in a way that might make it easier to compare them and solve for the scalar $t$ in the linear combination.

Non-collinear Points

Points are considered non-collinear when they do not all lie on a single straight line. In a two-dimensional plane, any two points are always collinear because you can draw a straight line through them. However, when there are three or more points, they may not all lie on the same line, and that is when we refer to them as non-collinear.

In order to establish whether points P, Q, and R from our problem are non-collinear, we test to see if any of the points can be expressed as a linear combination of the other two with the aforementioned conditions on the coefficients. If none can be, this implies that no single line can pass through all three points, confirming that they are non-collinear. Furthermore, if one can identify distinct slopes formed by pairs of points, that is also an indication of non-collinearity. Non-collinear points are the building blocks of most geometric shapes, as they allow for the creation of vertices that are not aligned on a single straight path.