issues is complex and dynamic. As society progresses, new ethical dilemmas and legal challenges will emerge, influenced
technological advances, evolving social norms, and changing cultural values. Whether in the realms of artificial intelligenc
social justice, intellectual property, or education, it is crucial to navigate these issues thoughtfully and with an awareness
their broader implications for society.Addressing ethical dilemmas often requires a careful balance between individual
rights and collective responsibilities, between innovation and fairness, and between the potential for progress and the ris
of harm. At the same time, legal frameworks must adapt to reflect evolving ethical principles, ensuring that laws serve th
public good and protect fundamental rights while fostering justice and equity.Moving forward,
Chapter 1 Solution
1.3
Given:F1= 4ˆi + 3ˆj and F2= 1ˆi + 7ˆj
Find: angle θ between F1 and F2
Solution: Definition of dot product eqn (1.16 page 10)
u · v = |u||v|cosθ = u1v1 + u2v2 + u3v3
So
F1 · F2 = |F1||F2|cosθ = F11F21 + F12F22 + F13F33
√
|F1| = 42 + 32 = 5
√ √
|F2| = 12 + 72 = 5 2
Therefore
√
5 × 5 2 × cosθ = 4 × 1 + 0 + 3 × 7
4×1+0+3×7 1
cosθ = √ = √
5×5 2 2
−1 1 ◦
θ = cos ( √ ) = 45 (ans)
2
, Given:F1= −5ˆi + 3ˆj and F2= 1ˆi − 4ˆj
Find:The cross product of F1 and F2
issues is complex and dynamic. As society progresses, new ethical dilemmas and legal challenge
will emerge, influenced by technological advances, evolving social norms, and changing cultura
values. Whether in the realms of artificial intelligence, social justice, intellectual property, or
education, it is crucial to navigate these issues thoughtfully and with an awareness of their
broader implications for society.Addressing ethical dilemmas often requires a careful balance
between individual rights and collective responsibilities, between innovation and fairness, and
between the potential for progress and the risk of harm. At the same time, legal frameworks
must adapt to reflect evolving ethical principles, ensuring that laws serve the public good and
protect fundamental rights while fostering justice and equity.Moving forward,
Solution: Definition of cross product (1.22 and 1.23 page 11)
ˆi ˆj ˆ
u × v = u1 u2 u3
= (u2v3 − u3v2)ˆi + (u3v1 − v 1 u 3 ) ĵ + (u1v2 − u 2 v 1 )k̂
So
F1 × F2 = (F12F23 − F13F22)ˆi + (F13F21 − F 11 F 23 ) ĵ + (F11F22 − F 12 F 21 )k̂
= (3 × 0 − 0 × (−4))ˆi + (0 × 1 − (−5) × 0)ˆj + ((−5) × (−4) − 3 × 1)kˆ
= 0ˆi + 0ˆj + 17kˆ (ans)
technological advances, evolving social norms, and changing cultural values. Whether in the realms of artificial intelligenc
social justice, intellectual property, or education, it is crucial to navigate these issues thoughtfully and with an awareness
their broader implications for society.Addressing ethical dilemmas often requires a careful balance between individual
rights and collective responsibilities, between innovation and fairness, and between the potential for progress and the ris
of harm. At the same time, legal frameworks must adapt to reflect evolving ethical principles, ensuring that laws serve th
public good and protect fundamental rights while fostering justice and equity.Moving forward,
Chapter 1 Solution
1.3
Given:F1= 4ˆi + 3ˆj and F2= 1ˆi + 7ˆj
Find: angle θ between F1 and F2
Solution: Definition of dot product eqn (1.16 page 10)
u · v = |u||v|cosθ = u1v1 + u2v2 + u3v3
So
F1 · F2 = |F1||F2|cosθ = F11F21 + F12F22 + F13F33
√
|F1| = 42 + 32 = 5
√ √
|F2| = 12 + 72 = 5 2
Therefore
√
5 × 5 2 × cosθ = 4 × 1 + 0 + 3 × 7
4×1+0+3×7 1
cosθ = √ = √
5×5 2 2
−1 1 ◦
θ = cos ( √ ) = 45 (ans)
2
, Given:F1= −5ˆi + 3ˆj and F2= 1ˆi − 4ˆj
Find:The cross product of F1 and F2
issues is complex and dynamic. As society progresses, new ethical dilemmas and legal challenge
will emerge, influenced by technological advances, evolving social norms, and changing cultura
values. Whether in the realms of artificial intelligence, social justice, intellectual property, or
education, it is crucial to navigate these issues thoughtfully and with an awareness of their
broader implications for society.Addressing ethical dilemmas often requires a careful balance
between individual rights and collective responsibilities, between innovation and fairness, and
between the potential for progress and the risk of harm. At the same time, legal frameworks
must adapt to reflect evolving ethical principles, ensuring that laws serve the public good and
protect fundamental rights while fostering justice and equity.Moving forward,
Solution: Definition of cross product (1.22 and 1.23 page 11)
ˆi ˆj ˆ
u × v = u1 u2 u3
= (u2v3 − u3v2)ˆi + (u3v1 − v 1 u 3 ) ĵ + (u1v2 − u 2 v 1 )k̂
So
F1 × F2 = (F12F23 − F13F22)ˆi + (F13F21 − F 11 F 23 ) ĵ + (F11F22 − F 12 F 21 )k̂
= (3 × 0 − 0 × (−4))ˆi + (0 × 1 − (−5) × 0)ˆj + ((−5) × (−4) − 3 × 1)kˆ
= 0ˆi + 0ˆj + 17kˆ (ans)
