1.1.19 Vector Algebra

Condition: the peaks of the pyramid \ (a b c d: \) 1. Build a pyramid in the Cartesian coordinate system, 2. Determine the coordinates and lengths of vectors \ (\ overrightarrow {a b}, \ overrightarrow {a c}, \ overrightarrow {a d} \). 3. Determine the direction \ (\ overrightarrow {a b} \) (its guide cosine). 4. Find \ ((\ overrightarrow {a b}; \ overrightarrow {a c}) \). 5. Find the projection of the vector \ (\ overrightarrow {a c} \) to the direction \ (\ overrightarrow {a b} \). 6. Calculate \ ([\ overrightarrow {a b}, \ overrightarrow {a c}] \). 7. Calculate \ ((\ overrightarrow {a b}-\ overrightarrow {a c})^{2} \). 8. Write out the equation of the plane passing through the points \ (a, b, C \). 9. Find the area of the line \ (a b c \). 10. Find the volume of the pyramid \ (a b c d \). 11. Write out the equation of heights lowered from the top \ (d \) to the line \ (a b c \). 12. Find the length of the height lowered from the \ (d \) point on the line \ (a b c \). \ (A (2; 3; 1), \ quad b (4; 1; -2), \ quad C (6; 3; 7), \ quad D (7; 5; -3) \).