By the end of this chapter, you will be able to: - confidently perform calculations with different types of numbers using the right mathematical concepts, - accurately carry out arithmetic operations on integers by applying the correct methods, - solve mathematical problems correctly by choosing and using the appropriate concepts.
Mastering these skills will help you tackle real-world engineering challenges with confidence and precision.
Vectors and scalars are fundamental in biomedical engineering, especially when analyzing forces, motions, and fields in physiological systems and medical devices. Biomedical engineers in Kenya often encounter problems involving two- and three-dimensional vectors, such as blood flow velocity, electrical signals, and mechanical stresses in prosthetics. Mastery of vector and scalar quantities enables accurate modeling and problem-solving in design, diagnostics, and treatment applications.
Vectors and scalars describe physical quantities that are critical in biomedical engineering. Scalars have only magnitude, such as temperature or mass, while vectors have both magnitude and direction, such as force or velocity. This section covers vector representation, operations, and applications in two and three dimensions relevant to biomedical contexts.
Scalars are quantities described fully by a magnitude alone and have no direction. Examples include temperature, mass, and time. Vectors require both magnitude and direction for complete description, such as displacement, velocity, and force.
The vector magnitude is denoted by \( |\mathbf{A}| \) and its direction by an angle or coordinate components. Vectors in two dimensions are often expressed as \(\mathbf{A} = A_x \mathbf{i} + A_y \mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors along the x- and y-axes respectively. In three dimensions, vectors are \(\mathbf{A} = A_x \mathbf{i} + A_y \mathbf{j} + A_z \mathbf{k}\), with \(\mathbf{k}\) along the z-axis.
$$|\mathbf{A}| = \sqrt{A_x^2 + A_y^2} \quad \text{(2D)}$$
$$|\mathbf{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2} \quad \text{(3D)}$$
Example 1: A sensor measures blood velocity as \(A_x = 3\, m/s\) and \(A_y = 4\, m/s\) in a plane. Find the velocity magnitude.
Given: \(A_x = 3\, m/s\), \(A_y = 4\, m/s\)
$$|\mathbf{A}| = \sqrt{A_x^2 + A_y^2}$$
$$|\mathbf{A}| = \sqrt{3^2 + 4^2}$$
$$|\mathbf{A}| = \sqrt{9 + 16}$$
$$|\mathbf{A}| = \sqrt{25}$$
Answer: 5\, m/s
Example 2: A force vector in 3D is given by components \(F_x = 5\, N\), \(F_y = 2\, N\), and \(F_z = 6\, N\). Calculate the resultant force magnitude.
Given: \(F_x = 5\, N\), \(F_y = 2\, N\), \(F_z = 6\, N\)
$$|\mathbf{F}| = \sqrt{F_x^2 + F_y^2 + F_z^2}$$
$$|\mathbf{F}| = \sqrt{5^2 + 2^2 + 6^2}$$
$$|\mathbf{F}| = \sqrt{25 + 4 + 36}$$
$$|\mathbf{F}| = \sqrt{65}$$
$$|\mathbf{F}| \approx 8.06\, N$$
Answer: 8.06\, N
Example 3: An electrode displacement vector is \(\mathbf{D} = -3\mathbf{i} + 4\mathbf{j} + 12\mathbf{k}\) mm. Find its magnitude.
Given: \(D_x = -3\, mm\), \(D_y = 4\, mm\), \(D_z = 12\, mm\)
$$|\mathbf{D}| = \sqrt{(-3)^2 + 4^2 + 12^2}$$
$$|\mathbf{D}| = \sqrt{9 + 16 + 144}$$
$$|\mathbf{D}| = \sqrt{169}$$
Answer: 13\, mm
Vectors combine according to their components. Addition and subtraction are performed component-wise. For vectors \(\mathbf{A}\) and \(\mathbf{B}\):
$$\mathbf{A} + \mathbf{B} = (A_x + B_x) \mathbf{i} + (A_y + B_y) \mathbf{j} + (A_z + B_z) \mathbf{k}$$
$$\mathbf{A} - \mathbf{B} = (A_x - B_x) \mathbf{i} + (A_y - B_y) \mathbf{j} + (A_z - B_z) \mathbf{k}$$
This is essential in biomedical engineering for combining forces, velocities, or fields from multiple sources.
Example 1: Two forces in a prosthetic limb joint are \(\mathbf{F}_1 = 3\mathbf{i} + 5\mathbf{j}\) N and \(\mathbf{F}_2 = -2\mathbf{i} + 4\mathbf{j}\) N. Find the resultant force.
Given: \(\mathbf{F}_1 = 3\mathbf{i} + 5\mathbf{j}\) N, \(\mathbf{F}_2 = -2\mathbf{i} + 4\mathbf{j}\) N
$$\mathbf{F}_R = \mathbf{F}_1 + \mathbf{F}_2 = (3 - 2)\mathbf{i} + (5 + 4)\mathbf{j}$$
$$\mathbf{F}_R = 1\mathbf{i} + 9\mathbf{j}$$
Answer: \(\mathbf{F}_R = 1\mathbf{i} + 9\mathbf{j}\) N
Example 2: A blood flow velocity vector is \(\mathbf{V}_1 = 2\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}\) m/s. Another flow vector is \(\mathbf{V}_2 = -1\mathbf{i} + 2\mathbf{j} + 1\mathbf{k}\) m/s. Calculate the net velocity vector.
Given: \(\mathbf{V}_1 = 2\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}\) m/s, \(\mathbf{V}_2 = -1\mathbf{i} + 2\mathbf{j} + 1\mathbf{k}\) m/s
$$\mathbf{V}_{net} = (2 - 1)\mathbf{i} + (3 + 2)\mathbf{j} + (4 + 1)\mathbf{k}$$
$$\mathbf{V}_{net} = 1\mathbf{i} + 5\mathbf{j} + 5\mathbf{k}$$
Answer: \(\mathbf{V}_{net} = 1\mathbf{i} + 5\mathbf{j} + 5\mathbf{k}\) m/s
Example 3: In a rehabilitation device, two displacement vectors are \(\mathbf{D}_1 = 4\mathbf{i} - 3\mathbf{j} + 2\mathbf{k}\) mm and \(\mathbf{D}_2 = -1\mathbf{i} + 6\mathbf{j} - 4\mathbf{k}\) mm. Find the difference \(\mathbf{D}_1 - \mathbf{D}_2\).
Given: \(\mathbf{D}_1 = 4\mathbf{i} - 3\mathbf{j} + 2\mathbf{k}\) mm, \(\mathbf{D}_2 = -1\mathbf{i} + 6\mathbf{j} - 4\mathbf{k}\) mm
$$\mathbf{D}_1 - \mathbf{D}_2 = (4 - (-1))\mathbf{i} + (-3 - 6)\mathbf{j} + (2 - (-4))\mathbf{k}$$
$$\mathbf{D}_1 - \mathbf{D}_2 = 5\mathbf{i} - 9\mathbf{j} + 6\mathbf{k}$$
Answer: \(\mathbf{D}_1 - \mathbf{D}_2 = 5\mathbf{i} - 9\mathbf{j} + 6\mathbf{k}\) mm
The scalar product of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) gives a scalar quantity defined as:
$$\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \theta = A_x B_x + A_y B_y + A_z B_z$$
where \(\theta\) is the angle between the vectors. This operation is useful in biomedical engineering to find work done by a force, projection of one vector onto another, or to determine angles between vectors like muscle force directions.
Example 1: Calculate the dot product of vectors \(\mathbf{A} = 3\mathbf{i} + 4\mathbf{j}\) and \(\mathbf{B} = 2\mathbf{i} + 5\mathbf{j}\).
Given: \(\mathbf{A} = 3\mathbf{i} + 4\mathbf{j}\), \(\mathbf{B} = 2\mathbf{i} + 5\mathbf{j}\)
$$\mathbf{A} \cdot \mathbf{B} = 3 \times 2 + 4 \times 5$$
$$\mathbf{A} \cdot \mathbf{B} = 6 + 20$$
Answer: 26
Example 2: Two force vectors in 3D: \(\mathbf{F}_1 = 1\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\) N and \(\mathbf{F}_2 = 4\mathbf{i} + 0\mathbf{j} - 1\mathbf{k}\) N. Find their dot product.
Given: \(\mathbf{F}_1 = 1\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\), \(\mathbf{F}_2 = 4\mathbf{i} + 0\mathbf{j} - 1\mathbf{k}\)
$$\mathbf{F}_1 \cdot \mathbf{F}_2 = 1 \times 4 + 2 \times 0 + 3 \times (-1)$$
$$\mathbf{F}_1 \cdot \mathbf{F}_2 = 4 + 0 - 3$$
Answer: 1\, N^2
Example 3: A muscle applies a force \(\mathbf{F} = 50\mathbf{i} + 30\mathbf{j}\) N at an angle. Calculate the work done if displacement is \(\mathbf{d} = 2\mathbf{i} + 3\mathbf{j}\) m.
Given: \(\mathbf{F} = 50\mathbf{i} + 30\mathbf{j}\) N, \(\mathbf{d} = 2\mathbf{i} + 3\mathbf{j}\) m
$$W = \mathbf{F} \cdot \mathbf{d} = 50 \times 2 + 30 \times 3$$
$$W = 100 + 90$$
Answer: 190\, J
The cross product of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) produces a vector perpendicular to both, defined by:
$$\mathbf{A} \times \mathbf{B} = (A_y B_z - A_z B_y) \mathbf{i} - (A_x B_z - A_z B_x) \mathbf{j} + (A_x B_y - A_y B_x) \mathbf{k}$$
The magnitude is:
$$|\mathbf{A} \times \mathbf{B}| = |\mathbf{A}| |\mathbf{B}| \sin \theta$$
This is important in biomedical engineering for calculating torque in joints, angular momentum of rotating devices, and electromagnetic force directions.
Example 1: Find the cross product of \(\mathbf{A} = 2\mathbf{i} + 3\mathbf{j}\) and \(\mathbf{B} = 4\mathbf{i} + \mathbf{j}\).
Given: \(\mathbf{A} = 2\mathbf{i} + 3\mathbf{j} + 0\mathbf{k}\), \(\mathbf{B} = 4\mathbf{i} + 1\mathbf{j} + 0\mathbf{k}\)
$$\mathbf{A} \times \mathbf{B} = (3 \times 0 - 0 \times 1)\mathbf{i} - (2 \times 0 - 0 \times 4)\mathbf{j} + (2 \times 1 - 3 \times 4)\mathbf{k}$$
$$\mathbf{A} \times \mathbf{B} = 0\mathbf{i} - 0\mathbf{j} + (2 - 12)\mathbf{k}$$
$$\mathbf{A} \times \mathbf{B} = -10\mathbf{k}$$
Answer: \(-10\mathbf{k}\)
Example 2: Calculate the torque vector produced by a force \(\mathbf{F} = 6\mathbf{i} + 3\mathbf{j} + 2\mathbf{k}\) N applied at position vector \(\mathbf{r} = 2\mathbf{i} + 0\mathbf{j} + 1\mathbf{k}\) m.
Given: \(\mathbf{F} = 6\mathbf{i} + 3\mathbf{j} + 2\mathbf{k}\), \(\mathbf{r} = 2\mathbf{i} + 0\mathbf{j} + 1\mathbf{k}\)
$$\mathbf{\tau} = \mathbf{r} \times \mathbf{F} = (0 \times 2 - 1 \times 3)\mathbf{i} - (2 \times 2 - 1 \times 6)\mathbf{j} + (2 \times 3 - 0 \times 6)\mathbf{k}$$
$$\mathbf{\tau} = (0 - 3)\mathbf{i} - (4 - 6)\mathbf{j} + (6 - 0)\mathbf{k}$$
$$\mathbf{\tau} = -3\mathbf{i} + 2\mathbf{j} + 6\mathbf{k}$$
Answer: \(-3\mathbf{i} + 2\mathbf{j} + 6\mathbf{k}\) N·m
Example 3: Two vectors \(\mathbf{A} = 1\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\) and \(\mathbf{B} = 4\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}\) represent magnetic field and current direction respectively. Calculate \(\mathbf{A} \times \mathbf{B}\).
Given: \(\mathbf{A} = 1\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\), \(\mathbf{B} = 4\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}\)
$$\mathbf{A} \times \mathbf{B} = (2 \times 6 - 3 \times 5)\mathbf{i} - (1 \times 6 - 3 \times 4)\mathbf{j} + (1 \times 5 - 2 \times 4)\mathbf{k}$$
$$\mathbf{A} \times \mathbf{B} = (12 - 15)\mathbf{i} - (6 - 12)\mathbf{j} + (5 - 8)\mathbf{k}$$
$$\mathbf{A} \times \mathbf{B} = -3\mathbf{i} + 6\mathbf{j} - 3\mathbf{k}$$
Answer: \(-3\mathbf{i} + 6\mathbf{j} - 3\mathbf{k}\)
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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 vector operations including addition and subtraction, demonstrating how these combine vectors graphically and algebraically. The concept of position vectors was explained to locate points in space relative to an origin, which is essential in engineering contexts. Techniques for resolving vectors into components were discussed, enabling the simplification of complex vector quantities into manageable parts. The chapter also highlighted methods for estimating values in engineering calculations to ensure practical and efficient problem-solving. Together, these topics provide a comprehensive foundation for applying vector theorems in various engineering scenarios.
A biomedical engineer needs to find the resultant force acting on a prosthetic limb joint. Two forces act at the joint: \( \mathbf{F_1} = 3\mathbf{i} + 4\mathbf{j} \, \text{N} \) and \( \mathbf{F_2} = -2\mathbf{i} + 5\mathbf{j} \, \text{N} \). Calculate the resultant force vector. (2 marks)
Determine the magnitude and direction of the vector \( \mathbf{A} = 6\mathbf{i} - 8\mathbf{j} \) representing the electrical field intensity in a biomedical sensor. (2 marks)
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