Question

The vectors \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) are coplanar and related by $$ \lambda \mathbf{a}+\mu \mathbf{b}+v \mathbf{c}=0 $$ where \(\lambda, \mu, v\) are not all zero. Show that the condition for the points with position vectors \(\alpha \mathbf{a}, \beta \mathbf{b}\) and \(\gamma \mathrm{c}\) to be collinear is $$ \frac{\lambda}{\alpha}+\frac{\mu}{\beta}+\frac{v}{\gamma}=0 $$

Short answer

The collinearity condition for the points is \(\frac{\lambda}{\alpha} + \frac{\mu}{\beta} + \frac{v}{\gamma} = 0\).

Step-by-step solution

- Understand the Co-planarity Condition

Given the vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) are co-planar, this implies there is a linear dependency among them. This is expressed as \(\lambda \mathbf{a} + \mu \mathbf{b} + v \mathbf{c} = 0\), where \(\lambda\), \(\mu\), and \(v\) are not all zero.

- Express Points' Position Vectors

The position vectors of the points are given as \(\alpha \mathbf{a}\), \(\beta \mathbf{b}\), and \(\gamma \mathbf{c}\). To be collinear, these points must satisfy a specific linear relationship.

- Set Up the Collinearity Condition

For the points \(\alpha \mathbf{a}\), \(\beta \mathbf{b}\), and \(\gamma \mathbf{c}\) to be collinear, there must exist scalars \(k_1\), \(k_2\), and \(k_3\) such that \(k_1 (\alpha \mathbf{a}) + k_2 (\beta \mathbf{b}) + k_3 (\gamma \mathbf{c}) = 0\). Since these are position vectors of collinear points, the coefficients \(k_1\), \(k_2\), and \(k_3\) must be related to \(\lambda\), \(\mu\), and \(v\).

- Relate Scalars to Given Coefficients

From the linear dependency, \(\lambda \mathbf{a} + \mu \mathbf{b} + v \mathbf{c} = 0\), we equate each vector component to zero for collinearity:\[k_1 (\alpha \mathbf{a}) + k_2 (\beta \mathbf{b}) + k_3 (\gamma \mathbf{c}) = 0\].By comparing, we equate the coefficients on both sides, leading us to \(k_1 \alpha = \lambda\), \(k_2 \beta = \mu\), \(k_3 \gamma = v\).

- Simplify the Equation

The equation \(\lambda \mathbf{a} + \mu \mathbf{b} + v \mathbf{c} = 0\) can be rewritten using the position vectors to determine the collinearity condition.From the equations above we have, \(k_1 = \frac{\lambda}{\alpha}\), \(k_2 = \frac{\mu}{\beta}\), and \(k_3 = \frac{v}{\gamma}\).

- Apply the Condition for Linear Dependence

Since the position vectors are linearly dependent, their scalars must satisfy:\[\frac{\lambda}{\alpha} + \frac{\mu}{\beta} + \frac{v}{\gamma} = 0\].This is the required condition for collinearity of points with position vectors \(\alpha \mathbf{a}\), \(\beta \mathbf{b}\), and \(\gamma \mathbf{c}\).

Key concepts

collinearity

Collinearity means that three or more points lie on the same straight line. For vectors, three points with position vectors \ \(\alpha \mathbf{a}\), \(\beta \mathbf{b}\), and \(\gamma \mathbf{c}\) are collinear if they satisfy a specific linear relationship.

To establish collinearity, we need to find scalars that make the vectors linearly dependent. In other words, \ \(\alpha \mathbf{a}\), \(\beta \mathbf{b}\), and \(\gamma \mathbf{c}\) are collinear if:

• \ \( k_1 (\alpha \mathbf{a}) + k_2 (\beta \mathbf{b}) + k_3 (\gamma \mathbf{c}) = 0 \)

This condition arises naturally from the property of collinear points and can be used to derive equations that test collinearity.

linear dependence

Linear dependence involves vectors that can be expressed as combinations of each other. If vectors \ \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) are linearly dependent, there exists scalars \ \(\lambda, \mu\), and \ \(v\) (not all zero) such that:
\ \( \lambda \mathbf{a} + \mu \mathbf{b} + v \mathbf{c} = 0 \)

In the context of the provided exercise, showing the collinearity condition boils down to understanding this linear dependence. When rewritten in the context of points' position vectors, the equation transforms into:

• \ \( \frac{\lambda}{\alpha} + \frac{\mu}{\beta} + \frac{v}{\gamma} = 0 \)
  
  

This indicates that the linear dependence of the original vectors \ \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) translates directly to the collinearity condition of the transformed vectors.

vector algebra

Vector algebra is fundamental for solving problems involving vectors. It includes operations like vector addition, scalar multiplication, and finding linear combinations.

For instance, in the given problem, the key equation

• \ \( \lambda \mathbf{a} + \mu \mathbf{b} + v \mathbf{c} = 0 \)

is a result of vector algebra.

Such equations can be manipulated using principles of vector addition and scalar multiplication to reveal important geometric properties.

When working with position vectors and testing for collinearity, using vector algebra makes it easy to break down the problem into manageable steps and derive meaningful conditions.

position vectors

Position vectors describe the location of points in space relative to a fixed origin. In this problem, the position vectors \ \(\alpha \mathbf{a}\), \(\beta \mathbf{b}\), and \(\gamma \mathbf{c}\) are given.
These vectors are key to representing the coordinates of the points in vector form.

By manipulating these position vectors, we can set up equations and conditions to explore relationships like collinearity:

• \ \( k_1 (\alpha \mathbf{a}) + k_2 (\beta \mathbf{b}) + k_3 (\gamma \mathbf{c}) = 0 \)

The simplicity and versatility of position vectors allow us to investigate complex geometric relationships using straightforward algebraic techniques.