3. Vectors

Using the vectors given in the figure, (a) find A ● B. (b) Find A ● C.

Find the scalar product of the vectors A and B given in Exercise 1.38.

What is the angle Φ between vectors E and F in FIGURE P3.24?

(III) The arrangement of atoms in zinc is an example of 'hexagonal close-packed' structure. Three of the nearest neighbors are found at the following (x, y, z) coordinates, given in nanometers (10⁻⁹m) : atom 1 is at (0, 0, 0); atom 2 is at (0.230, 0.133, 0); atom 3 is at (0.077, 0.133, 0.247). Find the angle between two vectors: one that connects atom 1 with atom 2 and another that connects atom 1 with atom 3.

(II) Let V→ = 20.0î + 26.0ĵ - 14.0k̂ . What angles does this vector make with the x, y, and z axes?

(II) A→ and B→ are two vectors in the xy plane that make angles a and ᵦ with the x axis respectively. Evaluate the scalar product of A→ and B→ and deduce the following trigonometric identity: cos ( a and ᵦ) = cos a cosᵦ + sin a sin ᵦ.

(I) Vector V₁→ points along the z axis and has magnitude V₁ = 75. Vector V₂→ lies in the xz plane, has magnitude V₂ = 48 ,and makes a -48° angle with the x axis (points below the x axis). What is the scalar product V₁→ • V₂→ ?

(II) Given vectors A→ = 4.8î + 6.8ĵ and B→ = 9.6î + 6.7ĵ , determine the vector C→ that lies in the xy plane, is perpendicular to B→ , and whose dot product with A→ is 18.0.

The treasure map in FIGURE P3.41 gives the following directions to the buried treasure: 'Start at the old oak tree, walk due north for 500 paces, then due east for 100 paces. Dig.' But when you arrive, you find an angry dragon just north of the tree. To avoid the dragon, you set off along the yellow brick road at an angle 60° east of north. After walking 300 paces you see an opening through the woods. In which direction should you walk, as an angle west of north, and how far, to reach the treasure?

Vector ${\vec{V}}_1$ points along the $z$ axis and has magnitude $V_1=75$. Vector ${\vec{V}}_2$ lies in the $xz$ plane, has magnitude $V_2=48$, and makes a $-{48}^{\circ }$ angle with the $x$ axis (points below the $x$ axis). What is the scalar product ${\vec{V}}_1\cdot {\vec{V}}_2$?

Given vectors $\vec{A}=-4.8\hat{i}+6.8\hat{j}$ and $\vec{B}=9.6\hat{i}+6.7\hat{j}$, determine the vector $\vec{C}$ that lies in the $xy$ plane, is perpendicular to $\vec{B}$, and whose dot product with $\vec{A}$ is 18.0.

$\overrightarrow{A}$ and $\overrightarrow{B}$ are two vectors in the $xy$ plane that make angles $\alpha$ and $\beta$ with the $x$ axis respectively. Evaluate the scalar product of $\overrightarrow{A}$ and $\overrightarrow{B}$ and deduce the following trigonometric identity: $\cos (\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta .$

The arrangement of atoms in zinc is an example of 'hexagonal close-packed' structure. Three of the nearest neighbors are found at the following (x, y, z) coordinates, given in nanometers (10^-9m): atom 1 is at (0, 0, 0); atom 2 is at (0.230, 0.133, 0); atom 3 is at (0.077, 0.133, 0.247). Find the angle between two vectors: one that connects atom 1 with atom 2 and another that connects atom 1 with atom 3.

(II) Let $\vec{V}=20.0\hat{i}+26.0\hat{j}-14.0\hat{k}$. What angles does this vector make with the x, y, and z axes?