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CIE A-Level Maths Study Notes

2.7.6 Scalar Product and Its Applications

The scalar product, also known as the dot product, is a pivotal concept in vector mathematics, playing a crucial role in understanding angles, projections, and spatial relationships in both 2D and 3D spaces. This comprehensive guide delves into its definition, intricate calculation methods, and a wide array of applications.

Definition and Calculation of Scalar Product

Scalar Product: For vectors a=aii+ajj+akk\mathbf{a} = a_i \mathbf{i} + a_j \mathbf{j} + a_k \mathbf{k} and b=bii+bjj+bkk\mathbf{b} = b_i \mathbf{i} + b_j \mathbf{j} + b_k \mathbf{k} in 3D space, the scalar product is ab=aibi+ajbj+akbk\mathbf{a} \cdot \mathbf{b} = a_i b_i + a_j b_j + a_k b_k.

Purpose: Measures how much two vectors align, indicating their directional similarity.

Scalar Product

Image courtesy of Cuemath

Calculating Angles Between Vectors

Formula:

  • Use cosθ=abab\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} to find the angle θ\theta between vectors a\mathbf{a} and b\mathbf{b}.

Application:

  • Essential for finding the exact angle between two vectors.
  • Identifies orthogonal vectors at 9090^\circ and parallel vectors at 00^\circ or 180)180^\circ ).
Angles between vectors

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Example:

Find the angle between two vectors a=3i+4j\mathbf{a} = 3\mathbf{i} + 4\mathbf{j} and b=4i+3j\mathbf{b} = -4\mathbf{i} + 3\mathbf{j}.

Solution:

1. Scalar Product:

ab=(3)(4)+(4)(3)=12+12=0\mathbf{a} \cdot \mathbf{b} = (3)(-4) + (4)(3) = -12 + 12 = 0

2. Magnitudes:

a=32+42=25=5|\mathbf{a}| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5b=(4)2+32=25=5|\mathbf{b}| = \sqrt{(-4)^2 + 3^2} = \sqrt{25} = 5

3. Angle Between Vectors:

Use cosθ=abab\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} leading to cosθ=025=0.\cos \theta = \frac{0}{25} = 0.

4. Determine θ\theta:

cosθ=0\cos \theta = 0 implies θ=90\theta = 90^\circ

Conclusion: The angle between a\mathbf{a} and b\mathbf{b} is 9090^\circ.

Perpendicularity and Projections

Scalar Product Uses:

  • Checks if vectors are orthogonal (perpendicular).
  • Calculates the shortest distance from a point to a plane or line.
  • Breaks down vectors into orthogonal components.

Importance:

  • Crucial in physics, engineering, and computer graphics.

Application in 3D Geometry

Scalar Product Role:

  • Essential for analyzing shapes like cuboids, pyramids, and tetrahedra.
  • Helps calculate surface areas, volumes, and angles between planes and lines.
  • Enhances understanding of spatial relationships and structures.
3D Geometry

Image courtesy of Calcworkshop

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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