Edexcel IAL - Pure Maths 4- 7.1 Vectors in 2D and 3D- Study notes - New syllabus
Edexcel IAL – Pure Maths 4- 7.1 Vectors in 2D and 3D -Study notes- New syllabus
Edexcel IAL – Pure Maths 4- 7.1 Vectors in 2D and 3D -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- 7.1 Vectors in 2D and 3D
Vectors in Two Dimensions
A vector is a quantity that has both magnitude and direction. In two dimensions, vectors are usually represented using Cartesian coordinates.
A vector \( \mathbf{a} \) can be written as:
\( \mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} \) or \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \)
Position Vectors
The position vector of a point \( A(x,y) \) is:
\( \vec{OA} = x\mathbf{i} + y\mathbf{j} \)
Magnitude of a Vector
The magnitude (length) of a vector \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \) is:
\( |\mathbf{a}| = \sqrt{a_1^2 + a_2^2} \)
Unit Vectors
A unit vector in the direction of \( \mathbf{a} \) is:
\( \dfrac{\mathbf{a}}{|\mathbf{a}|} \)
Vector Addition and Subtraction
If:
\( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \), \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} \)
Then:
\( \mathbf{a} + \mathbf{b} = (a_1+b_1)\mathbf{i} + (a_2+b_2)\mathbf{j} \)
\( \mathbf{a} – \mathbf{b} = (a_1-b_1)\mathbf{i} + (a_2-b_2)\mathbf{j} \)
Scalar Multiplication
If \( k \) is a scalar:
\( k\mathbf{a} = ka_1\mathbf{i} + ka_2\mathbf{j} \)
Example
Find the magnitude of \( \mathbf{a} = 3\mathbf{i} + 4\mathbf{j} \).
▶️ Answer / Explanation
\( |\mathbf{a}| = \sqrt{3^2 + 4^2} = \sqrt{9+16} = 5 \)
Example
Find the vector from \( A(2,3) \) to \( B(5,7) \).
▶️ Answer / Explanation
\( \vec{AB} = (5-2)\mathbf{i} + (7-3)\mathbf{j} = 3\mathbf{i} + 4\mathbf{j} \)
Example
Find a unit vector in the direction of \( \mathbf{a} = 6\mathbf{i} – 8\mathbf{j} \).
▶️ Answer / Explanation
\( |\mathbf{a}| = \sqrt{36 + 64} = 10 \)
Unit vector \( = \dfrac{6\mathbf{i}-8\mathbf{j}}{10} = \dfrac{3}{5}\mathbf{i} – \dfrac{4}{5}\mathbf{j} \)
Vectors in Three Dimensions
A vector in three dimensions has both magnitude and direction and is written using three components.
\( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \)
or as a column vector:
\( \mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} \)
Position Vectors
The position vector of a point \( A(x,y,z) \) is:
\( \vec{OA} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \)
Magnitude of a Vector
If:
\( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \)
then:
\( |\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \)
Vector Between Two Points
If:
\( A(x_1,y_1,z_1),\ B(x_2,y_2,z_2) \)
then:
\( \vec{AB} = (x_2-x_1)\mathbf{i} + (y_2-y_1)\mathbf{j} + (z_2-z_1)\mathbf{k} \)
Unit Vector
A unit vector in the direction of \( \mathbf{a} \) is:
\( \dfrac{\mathbf{a}}{|\mathbf{a}|} \)
Vector Addition and Scalar Multiplication
If:
\( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \), \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \)
then:
\( \mathbf{a} + \mathbf{b} = (a_1+b_1)\mathbf{i} + (a_2+b_2)\mathbf{j} + (a_3+b_3)\mathbf{k} \)
\( k\mathbf{a} = ka_1\mathbf{i} + ka_2\mathbf{j} + ka_3\mathbf{k} \)
Example
Find the magnitude of \( \mathbf{a} = 2\mathbf{i} – 3\mathbf{j} + 6\mathbf{k} \).
▶️ Answer / Explanation
\( |\mathbf{a}| = \sqrt{2^2 + (-3)^2 + 6^2} = \sqrt{4+9+36} = 7 \)
Example
Find the vector from \( A(1,2,3) \) to \( B(5,4,-1) \).
▶️ Answer / Explanation
\( \vec{AB} = (4)\mathbf{i} + (2)\mathbf{j} – 4\mathbf{k} \)
Example
Find a unit vector in the direction of \( \mathbf{a} = 3\mathbf{i} – 6\mathbf{j} + 2\mathbf{k} \).
▶️ Answer / Explanation
\( |\mathbf{a}| = \sqrt{9 + 36 + 4} = 7 \)
Unit vector \( = \dfrac{3\mathbf{i} – 6\mathbf{j} + 2\mathbf{k}}{7} \)