3. Vectors

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

(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.

(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.

(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.

(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.