If \({\rm{\vec b}}\) and \({\rm{\vec c}}\) are the position vectors of the points B and C respectively, then the position vector of the point D such that \(\overrightarrow {{\rm{BD}}} = 4{\rm{\;}}\overrightarrow {{\rm{BC}}} \) is
Finding Position Vector D
This problem asks us to find the position vector of a point D based on the position vectors of points B and C and a given vector relationship. We are given that \({\rm{\vec b}}\) is the position vector of B and \({\rm{\vec c}}\) is the position vector of C. This means that if O is the origin, \(\overrightarrow{{\rm{OB}}} = {\rm{\vec b}}\) and \(\overrightarrow{{\rm{OC}}} = {\rm{\vec c}}\).
Let the position vector of point D be \({\rm{\vec d}}\). This means \(\overrightarrow{{\rm{OD}}} = {\rm{\vec d}}\).
Understanding Vectors Between Points
A vector connecting two points, say from point P to point Q, denoted as \(\overrightarrow{{\rm{PQ}}}\), can be found by subtracting the position vector of the starting point (P) from the position vector of the ending point (Q).
- Vector from B to D: \(\overrightarrow{{\rm{BD}}} = \text{Position vector of D} - \text{Position vector of B} = {\rm{\vec d}} - {\rm{\vec b}}\)
- Vector from B to C: \(\overrightarrow{{\rm{BC}}} = \text{Position vector of C} - \text{Position vector of B} = {\rm{\vec c}} - {\rm{\vec b}}\)
Using the Given Vector Relationship
We are given the relationship: \(\overrightarrow {{\rm{BD}}} = 4{\rm{\;}}\overrightarrow {{\rm{BC}}}\).
Now, we substitute the expressions for \(\overrightarrow{{\rm{BD}}}\) and \(\overrightarrow{{\rm{BC}}}\) that we found using the position vectors:
\({\rm{\vec d}} - {\rm{\vec b}} = 4 \left( {\rm{\vec c}} - {\rm{\vec b}} \right)\)
Solving for the Position Vector of D
Our goal is to find \({\rm{\vec d}}\). Let's simplify the equation and isolate \({\rm{\vec d}}\):
First, distribute the 4 on the right side of the equation:
\({\rm{\vec d}} - {\rm{\vec b}} = 4{\rm{\vec c}} - 4{\rm{\vec b}}\)
Now, add \({\rm{\vec b}}\) to both sides of the equation to solve for \({\rm{\vec d}}\):
\({\rm{\vec d}} = 4{\rm{\vec c}} - 4{\rm{\vec b}} + {\rm{\vec b}}\)
Combine the \({\rm{\vec b}}\) terms:
\({\rm{\vec d}} = 4{\rm{\vec c}} + \left( -4 + 1 \right){\rm{\vec b}}\)
\({\rm{\vec d}} = 4{\rm{\vec c}} - 3{\rm{\vec b}}\)
So, the position vector of point D is \(4{\rm{\vec c}} - 3{\rm{\vec b}}\).
Comparing with Options
Let's compare our result with the given options:
- Option 1: \(4\left( {{\rm{\vec c}} - {\rm{\vec b}}} \right) = 4{\rm{\vec c}} - 4{\rm{\vec b}}\) (Incorrect)
- Option 2: \(- 4\left( {{\rm{\vec c}} - {\rm{\vec b}}} \right) = -4{\rm{\vec c}} + 4{\rm{\vec b}}\) (Incorrect)
- Option 3: \(4{\rm{\vec c}} - 3{\rm{\vec b}}\) (Correct)
- Option 4: \(4{\rm{\vec c}} + 3{\rm{\vec b}}\) (Incorrect)
Our calculated position vector for D matches Option 3.
| Concept | Formula |
|---|---|
| Position vector of point P | \(\overrightarrow{{\rm{OP}}} = {\rm{\vec p}}\) |
| Vector from P to Q | \(\overrightarrow{{\rm{PQ}}} = {\rm{\vec q}} - {\rm{\vec p}}\) |
| Given relationship | \(\overrightarrow {{\rm{BD}}} = 4{\rm{\;}}\overrightarrow {{\rm{BC}}}\) |
| Substituting vectors | \({\rm{\vec d}} - {\rm{\vec b}} = 4({\rm{\vec c}} - {\rm{\vec b}})\) |
| Position vector of D (\({\rm{\vec d}}}\)) | \(4{\rm{\vec c}} - 3{\rm{\vec b}}\) |
Revision Table: Position Vectors and Vector Operations
| Term | Definition/Concept | Notation |
|---|---|---|
| Position Vector | A vector that points from the origin (O) to a point (P). It uniquely identifies the location of the point in space relative to the origin. | \(\overrightarrow{{\rm{OP}}}\) or \({\rm{\vec p}}\) |
| Vector between two points | The vector from point A to point B is found by subtracting the position vector of A from the position vector of B. | \(\overrightarrow{{\rm{AB}}} = {\rm{\vec b}} - {\rm{\vec a}}\) |
| Scalar Multiplication of a Vector | Multiplying a vector by a scalar changes its magnitude (length). If the scalar is positive, the direction remains the same; if negative, the direction reverses. | \(k{\rm{\vec v}}\) |
| Vector Addition/Subtraction | Vectors can be added or subtracted component-wise, or geometrically using the triangle or parallelogram rule. In terms of position vectors, subtraction gives the vector connecting the points. | \({\rm{\vec a}} + {\rm{\vec b}}\), \({\rm{\vec a}} - {\rm{\vec b}}\) |
Additional Information: Geometric Interpretation
The relationship \(\overrightarrow {{\rm{BD}}} = 4{\rm{\;}}\overrightarrow {{\rm{BC}}}\) tells us two things:
- The vector \(\overrightarrow{{\rm{BD}}}\) is in the same direction as the vector \(\overrightarrow{{\rm{BC}}}\) because the scalar multiple is positive (4).
- The magnitude (length) of \(\overrightarrow{{\rm{BD}}}\) is 4 times the magnitude of \(\overrightarrow{{\rm{BC}}}\).
This means that points B, C, and D are collinear (lie on the same straight line). Furthermore, since \(\overrightarrow{{\rm{BD}}}\) is 4 times \(\overrightarrow{{\rm{BC}}}\), C lies between B and D, and the distance BD is four times the distance BC. You can think of this as C being on the line segment BD, such that the ratio of the lengths BC to CD is 1:3 (since BD = BC + CD = 4 BC, CD = 3 BC).
The position vector formula we derived, \({\rm{\vec d}} = 4{\rm{\vec c}} - 3{\rm{\vec b}}\), correctly reflects this relationship in terms of position vectors originating from the origin O. This formula can be rearranged as \({\rm{\vec d}} - {\rm{\vec c}} = 3{\rm{\vec c}} - 3{\rm{\vec b}} = 3({\rm{\vec c}} - {\rm{\vec b}})\), which means \(\overrightarrow{{\rm{CD}}} = 3\overrightarrow{{\rm{BC}}}\). This confirms that C is between B and D, and CD is 3 times BC, fitting the geometric picture derived from \(\overrightarrow {{\rm{BD}}} = 4{\rm{\;}}\overrightarrow {{\rm{BC}}}\).
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