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Vectors and scalars are fundamental in biomedical engineering calculations, especially when dealing with forces, velocities, and fields in two and three dimensions. Understanding how to represent and manipulate these quantities allows biomedical engineers to analyze biomechanical systems, medical imaging data, and physiological signals accurately. This chapter begins by exploring the mathematical representation of vectors and scalars in two and three-dimensional spaces, which forms the foundation for applying vector theorems in biomedical applications such as prosthetics design, biomechanics, and imaging technology.
Vectors in two dimensions are quantities having both magnitude and direction, often represented graphically as arrows or algebraically as ordered pairs. Scalars are quantities with magnitude only, such as temperature or mass. In biomedical engineering, two-dimensional vectors arise in analyzing planar motion of limbs or forces on joints.
A vector \(\mathbf{v}\) in two dimensions is usually written as
$$\mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j}$$
where \(v_x\) and \(v_y\) are the components along the \(x\) and \(y\) axes, respectively, and \(\mathbf{i}\), \(\mathbf{j}\) are the unit vectors along these axes.
The magnitude \(|\mathbf{v}|\) of vector \(\mathbf{v}\) is given by
$$|\mathbf{v}| = \sqrt{v_x^2 + v_y^2}$$
Example 1: A force vector acting on a knee joint has components \(v_x = 30\, \text{N}\) and \(v_y = 40\, \text{N}\). Find the magnitude of this force.
Given: \(v_x = 30\, \text{N}\), \(v_y = 40\, \text{N}\)
$$|\mathbf{v}| = \sqrt{v_x^2 + v_y^2}$$
$$|\mathbf{v}| = \sqrt{30^2 + 40^2}$$
$$|\mathbf{v}| = \sqrt{900 + 1600}$$
$$|\mathbf{v}| = \sqrt{2500}$$
Answer: 50 N
Example 2: A blood flow velocity vector in a 2D plane is \(v_x = 5\, \text{m/s}\), \(v_y = 12\, \text{m/s}\). Find the velocity magnitude.
Given: \(v_x = 5\, \text{m/s}\), \(v_y = 12\, \text{m/s}\)
$$|\mathbf{v}| = \sqrt{5^2 + 12^2}$$
$$|\mathbf{v}| = \sqrt{25 + 144}$$
$$|\mathbf{v}| = \sqrt{169}$$
Answer: 13 m/s
Example 3: A force vector in a planar prosthetic limb has components \(v_x = -15\, \text{N}\), \(v_y = 20\, \text{N}\). Find the magnitude and direction angle \(\theta\) measured from the positive \(x\)-axis.
Given: \(v_x = -15\, \text{N}\), \(v_y = 20\, \text{N}\)
Magnitude:
$$|\mathbf{v}| = \sqrt{(-15)^2 + 20^2}$$
$$|\mathbf{v}| = \sqrt{225 + 400}$$
$$|\mathbf{v}| = \sqrt{625} = 25\, \text{N}$$
Direction:
$$\theta = \tan^{-1} (\frac{v_y}{v_x}) = \tan^{-1} (\frac{20}{-15})$$
$$\theta = \tan^{-1}(-1.3333) = -53.13^\circ$$
Since \(v_x\) is negative and \(v_y\) positive, vector lies in the second quadrant:
$$\theta = 180^\circ - 53.13^\circ = 126.87^\circ$$
Answer: Magnitude = 25 N, Direction = 126.87 degrees
Biomedical engineering problems often require three-dimensional analysis, such as forces on implants, 3D blood flow, or electrical fields in tissues. A vector \(\mathbf{v}\) in three dimensions is expressed as
$$\mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k}$$
where \(v_x\), \(v_y\), and \(v_z\) are components along the \(x\), \(y\), and \(z\) axes respectively, and \(\mathbf{i}\), \(\mathbf{j}\), \(\mathbf{k}\) are the unit vectors.
The magnitude is
$$|\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$$
Example 1: A force vector acting on a hip implant has components \(v_x = 25\, \text{N}\), \(v_y = 30\, \text{N}\), and \(v_z = 10\, \text{N}\). Calculate the magnitude.
Given: \(v_x = 25\, \text{N}\), \(v_y = 30\, \text{N}\), \(v_z = 10\, \text{N}\)
$$|\mathbf{v}| = \sqrt{25^2 + 30^2 + 10^2}$$
$$|\mathbf{v}| = \sqrt{625 + 900 + 100}$$
$$|\mathbf{v}| = \sqrt{1625}$$
$$|\mathbf{v}| = 40.31\, \text{N}$$
Answer: 40.31 N
Example 2: The velocity of a fluid at a point inside an artery is given by \(v_x = 2\, \text{m/s}\), \(v_y = 3\, \text{m/s}\), \(v_z = 6\, \text{m/s}\). Find the speed.
Given: \(v_x = 2\, \text{m/s}\), \(v_y = 3\, \text{m/s}\), \(v_z = 6\, \text{m/s}\)
$$|\mathbf{v}| = \sqrt{2^2 + 3^2 + 6^2}$$
$$|\mathbf{v}| = \sqrt{4 + 9 + 36}$$
$$|\mathbf{v}| = \sqrt{49} = 7\, \text{m/s}$$
Answer: 7 m/s
Example 3: A resultant force vector on a biomedical sensor has components \(v_x = -10\, \text{N}\), \(v_y = 15\, \text{N}\), and \(v_z = -5\, \text{N}\). Find its magnitude.
Given: \(v_x = -10\, \text{N}\), \(v_y = 15\, \text{N}\), \(v_z = -5\, \text{N}\)
$$|\mathbf{v}| = \sqrt{(-10)^2 + 15^2 + (-5)^2}$$
$$|\mathbf{v}| = \sqrt{100 + 225 + 25}$$
$$|\mathbf{v}| = \sqrt{350}$$
$$|\mathbf{v}| = 18.71\, \text{N}$$
Answer: 18.71 N
Adding and subtracting vectors is essential for combining forces or displacements in biomedical engineering systems such as joint loading or electrical signal propagation.
Vector addition in components is done by adding corresponding components:
$$\mathbf{a} + \mathbf{b} = (a_x + b_x)\mathbf{i} + (a_y + b_y)\mathbf{j} + (a_z + b_z)\mathbf{k}$$
Subtraction is similar:
$$\mathbf{a} - \mathbf{b} = (a_x - b_x)\mathbf{i} + (a_y - b_y)\mathbf{j} + (a_z - b_z)\mathbf{k}$$
Example 1: Add vectors \(\mathbf{a} = 3\mathbf{i} + 4\mathbf{j}\) and \(\mathbf{b} = 1\mathbf{i} + 2\mathbf{j}\).
Given: \(\mathbf{a} = (3,4)\), \(\mathbf{b} = (1,2)\)
$$\mathbf{a} + \mathbf{b} = (3 + 1)\mathbf{i} + (4 + 2)\mathbf{j}$$
$$\mathbf{a} + \mathbf{b} = 4\mathbf{i} + 6\mathbf{j}$$
Answer: \(4\mathbf{i} + 6\mathbf{j}\)
Example 2: Subtract vectors \(\mathbf{a} = 5\mathbf{i} + 7\mathbf{j} + 2\mathbf{k}\) and \(\mathbf{b} = 3\mathbf{i} + 4\mathbf{j} + 1\mathbf{k}\).
Given: \(\mathbf{a} = (5,7,2)\), \(\mathbf{b} = (3,4,1)\)
$$\mathbf{a} - \mathbf{b} = (5 - 3)\mathbf{i} + (7 - 4)\mathbf{j} + (2 - 1)\mathbf{k}$$
$$\mathbf{a} - \mathbf{b} = 2\mathbf{i} + 3\mathbf{j} + 1\mathbf{k}$$
Answer: \(2\mathbf{i} + 3\mathbf{j} + 1\mathbf{k}\)
Example 3: A muscle force vector \(\mathbf{F_1} = 10\mathbf{i} + 15\mathbf{j} + 20\mathbf{k}\) N combines with \(\mathbf{F_2} = -5\mathbf{i} + 8\mathbf{j} + 12\mathbf{k}\) N. Find the resultant force.
Given: \(\mathbf{F_1} = (10,15,20)\) N, \(\mathbf{F_2} = (-5,8,12)\) N
$$\mathbf{R} = \mathbf{F_1} + \mathbf{F_2} = (10 - 5)\mathbf{i} + (15 + 8)\mathbf{j} + (20 + 12)\mathbf{k}$$
$$\mathbf{R} = 5\mathbf{i} + 23\mathbf{j} + 32\mathbf{k}$$
Answer: \(5\mathbf{i} + 23\mathbf{j} + 32\mathbf{k}\) N
Scalar multiplication scales a vector without changing its direction, which is important in biomedical engineering when adjusting force magnitudes or velocities. Unit vectors have magnitude 1 and indicate direction only.
Scalar multiplication of vector \(\mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k}\) by scalar \(k\) is
$$k \mathbf{v} = (k v_x) \mathbf{i} + (k v_y) \mathbf{j} + (k v_z) \mathbf{k}$$
A unit vector \(\hat{\mathbf{v}}\) in the direction of \(\mathbf{v}\) is
$$\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|} = \frac{v_x}{|\mathbf{v}|} \mathbf{i} + \frac{v_y}{|\mathbf{v}|} \mathbf{j} + \frac{v_z}{|\mathbf{v}|} \mathbf{k}$$
Example 1: Multiply vector \(\mathbf{v} = 4\mathbf{i} + 3\mathbf{j}\) by scalar 2.
Given: \(\mathbf{v} = (4,3)\), \(k=2\)
$$k \mathbf{v} = (2 \times 4) \mathbf{i} + (2 \times 3) \mathbf{j}$$
$$k \mathbf{v} = 8\mathbf{i} + 6\mathbf{j}$$
Answer: \(8\mathbf{i} + 6\mathbf{j}\)
Example 2: Find the unit vector in the direction of \(\mathbf{v} = 3\mathbf{i} + 4\mathbf{j}\).
Given: \(\mathbf{v} = (3,4)\)
Calculate magnitude:
$$|\mathbf{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$
Unit vector:
$$\hat{\mathbf{v}} = \frac{3}{5} \mathbf{i} + \frac{4}{5} \mathbf{j} = 0.6 \mathbf{i} + 0.8 \mathbf{j}$$
Answer: \(0.6 \mathbf{i} + 0.8 \mathbf{j}\)
Example 3: Scale vector \(\mathbf{v} = 5\mathbf{i} - 2\mathbf{j} + 4\mathbf{k}\) by scalar \(-3\).
Given: \(\mathbf{v} = (5, -2, 4)\), \(k = -3\)
$$k \mathbf{v} = (-3 \times 5) \mathbf{i} + (-3 \times -2) \mathbf{j} + (-3 \times 4) \mathbf{k}$$
$$k \mathbf{v} = -15 \mathbf{i} + 6 \mathbf{j} - 12 \mathbf{k}$$
Answer: \(-15 \mathbf{i} + 6 \mathbf{j} - 12 \mathbf{k}\)
Example 4: Find the unit vector of \(\mathbf{v} = 1\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}\).
Given: \(\mathbf{v} = (1,2,2)\)
Magnitude:
$$|\mathbf{v}| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3$$
Unit vector:
$$\hat{\mathbf{v}} = \frac{1}{3} \mathbf{i} + \frac{2}{3} \mathbf{j} + \frac{2}{3} \mathbf{k} = 0.333 \mathbf{i} + 0.667 \mathbf{j} + 0.667 \mathbf{k}$$
Answer: \(0.333 \mathbf{i} + 0.667 \mathbf{j} + 0.667 \mathbf{k}\)
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Create a free accountThis chapter introduced the fundamental concepts of vectors and scalars in both two and three dimensions, emphasizing their distinct characteristics and applications. It covered operations on vectors, focusing on how to perform addition and subtraction accurately to combine or compare vector quantities. The concept of position vectors was explored, showing how they represent the location of points in space relative to an origin. The chapter then examined the resolution of vectors into components along specified directions, which is essential for analyzing forces and movements in engineering problems. Finally, it detailed the scalar product and vector product, explaining their calculation methods and significance in determining magnitudes, directions, and areas related to vectors. Together, these topics provide a comprehensive foundation for applying vector theorems in practical engineering contexts.
Find the magnitude of the vector \(\vec{A} = 3\hat{i} + 4\hat{j}\) representing a displacement in a biomedical sensor alignment. (2 marks)
Calculate the resultant vector when \(\vec{B} = 5\hat{i} + 2\hat{j}\) and \(\vec{C} = -3\hat{i} + 6\hat{j}\) are added, representing forces applied on a prosthetic limb joint. (3 marks)
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