• :00Days
  • :00Hours
  • :00Mins
  • 00Seconds
A new era for learning is coming soonSign up for free
Log In Start studying!

Select your language

Suggested languages for you:
Answers without the blur. Sign up and see all textbooks for free! Illustration


Fundamentals Of Physics
Found in: Page 60

Answers without the blur.

Just sign up for free and you're in.


Short Answer

Here are three displacements, each measured in meters: d1=4.0i+5.0j-6.0k,d2=-1.0i+2.0j+3.0k ,andd3=4.0i+3.0j+2.0k. (a) What is r=d1+d2+d3 ? (b) What is the angle betweenr and the positive z axis? (c) What is the component of d1 along the direction of d2 (d) What is the component of d1 that is perpendicular to the direction of d2 and in the plane of role="math" localid="1658465314757" d1and d2 (Hint: For (c), consider Eq 3-20. and Fig, 3-18; for (d), consider Eq.3-24.)

a) The displacement r is 9.0i+6.0j-7.0k m

b) The angle betweenrand the positive z-axis is θ=123°

c) The component of d1 along the direction of role="math" localid="1658465562638" d2 is=-3.2m.

d) The component of d2 in the perpendicular direction of d2 is 8.2m.

See the step by step solution

Step by Step Solution

Step 1: Given data

The given vectors are,




Step 2: Understanding the concept

Vector addition and subtraction can be used to find the displacement and magnitude can be found using the formula. To find the angle between r and positive z axis we can use a scalar product of two vectors. Unit vector along z axis is. The components of d1can be found by using the property of the scalar product of two vectors.




Step 3: (a) Calculate the displacement r→

From equation (i), we can calculate the displacement vector r as follows.

r=4.0i+5.0j-6.0k--1.0i+2.0j+3.0k+4.0i+3.0j-2.0k =4.0i+5.0j-6.0k+1.0i-2.0j-3.0k+4.0i+3.0j+2.0k =9.0i+6.0j-7.0k

Therefore, the displacementr is9.0i+6.0j-7.0k.

Step 4: (b) Calculate the angle betweenr→and positive z axis

The magnitude of r is calculated as,

r=9.02+6.02+-7.02 =12.9 m

The angle betweenrand positive z axis is θ

localid="1658466997778" cosθ=r.kr =-7.012.0 θ=cos-1-0.543 =123°

The angle betweenlocalid="1658467048657" rand positive z axis is 123°.

Step 5: (c) Calculate the component of d→1 along the direction of d→2 

The component of d1along the direction of d1=d||

d=d1.u =d1cosϕ.............iii

u is a unit vector

d1.d2=d1d2cosϕ cosϕ=d1d2d1d2.............iv

Equation (iii) becomes as

d=d1cosϕ =d1d1.d2d1d2 =d1.d2d2

Now, calculate the dot product d1.d2 as,

role="math" localid="1658467900456" d1.d2=d1xd2x+d1yd2y+d1z+d2z =4.0-1.0+5.02.0+-6.03.0

Substitute it in the above equation.

role="math" localid="1658468009605" d=4.0-1.0+5.02.0+-6.03.0-1.022.0+3.02 =-3.2m

Step 6: (d) Calculate the component of d→1 in the perpendicular direction of d→2

The component of d1in the perpendicular direction of d2=d

d1=d+dd=d1-dd2=d12-dd=4.02+5.02+-6.02--3.22 =8.2 m

Therefore, the component of d1 in the perpendicular direction of d2is 8.2 m

Most popular questions for Physics Textbooks


Want to see more solutions like these?

Sign up for free to discover our expert answers
Get Started - It’s free

Recommended explanations on Physics Textbooks

94% of StudySmarter users get better grades.

Sign up for free
94% of StudySmarter users get better grades.