Figure P 3.28 illustrates typical proportions of male (m)...

Figure $\mathrm{P} 3.28$ illustrates typical proportions of male $(\mathrm{m})$ and female $(\mathrm{f})$ anatomies. The displacements $\overrightarrow{\mathrm{d}}_{1 \mathrm{~m}}$ and $\overrightarrow{\mathrm{d}}_{1 \mathrm{f}}$ from the soles of the feet to the navel have magnitudes of $104 \mathrm{~cm}$ and $84.0 \mathrm{~cm},$ respectively. The displacements $\overrightarrow{\mathbf{d}}_{2 \mathrm{~m}}$ and $\overrightarrow{\mathrm{d}}_{2 \mathrm{f}}$ from the navel to outstretched fingertips have magnitudes of $100 \mathrm{~cm}$ and $86.0 \mathrm{~cm},$ respectively. Find the vector sum of these displacements $\overrightarrow{\mathbf{d}}_{3}=\overrightarrow{\mathbf{d}}_{1}+\overrightarrow{\mathbf{d}}_{2}$ for both people.

Key Concepts

Vector Addition

Vector addition is the process of combining two or more vectors to form a resultant vector. This operation is performed by summing corresponding components of the given vectors, taking into account both their magnitudes and directions. It is fundamental in many areas of physics and mathematics for determining net effects from multiple vector quantities.

Displacement Vectors

A displacement vector describes a change in an object's position. It is characterized by both a magnitude and a direction, which distinguishes it from scalar quantities like distance. Displacement vectors are used to represent movement from one point to another and are central to understanding concepts in kinematics and mechanics.

Transcript

00:01 For this problem on the topic of vectors, we are given a figure which illustrates the typical proportions of males and females.

00:09 The displacements from the soles of the feet to the navel are given, and the displacements from the navel to the outstretched fingertips are also given for both males and females.

00:18 We want to find the vector sum of these displacement for both people.

00:25 So firstly, if we sum the components of the two vectors for the male, we get vector the 3m.

00:34 So we resolve this into components and find each of the components separately.

00:38 So d3mx is equal to d1mx plus d2mx where the subscript m stands for male and this is equal to 0 plus 100 centimeters times the cosine of 23 degrees.

01:11 This gives us d3 in the x direction to be 92 .1 centimeters or the x component of d3 for the male.

01:24 Now the y component of d3, d3 my similarly is d1mi plus d2.

01:37 So this is equal to 104 centimeters plus 100 centimeters times the sign of 23 degrees, which gives us d3my to be 143 .1 centimeters.

02:02 So we found the x and y components of d3 for the male.

02:06 Hence we can find the magnitude of d3, or d3m, is simply the square root of the sum of squares of each of these components.

02:18 That's a square root of 92 .1 centimeters all squared plus 143 .1 centimeters all squared.

02:35 So calculating we get d3 for the male to be 170 .1 centimeters, but we still have to find the direction...

Please give Ace some feedback