answer:1. To find the area of the parallelogram formed by vectors overrightarrow{AB} and overrightarrow{AC}, we first calculate these vectors: overrightarrow{AB} = B - A = (1 - (-1), 2 - 2, 1 - 1) = (2, 0, 0) overrightarrow{AC} = C - A = (-1 - (-1), 6 - 2, 4 - 1) = (0, 4, 3) Since the dot product of overrightarrow{AB} and overrightarrow{AC} is zero: overrightarrow{AB} cdot overrightarrow{AC} = (2, 0, 0) cdot (0, 4, 3) = 2 cdot 0 + 0 cdot 4 + 0 cdot 3 = 0 we conclude that overrightarrow{AB} is perpendicular to overrightarrow{AC}. Next, we find the magnitude of both vectors: |overrightarrow{AB}| = sqrt{2^2+0^2+0^2} = 2 |overrightarrow{AC}| = sqrt{0^2+4^2+3^2} = 5 The area S of the parallelogram is then the product of the magnitudes of these perpendicular vectors: S = |overrightarrow{AB}| cdot |overrightarrow{AC}| = 2 times 5 boxed{S = 10} 2. Let vector overrightarrow{a} be represented by (x, y, z). Since overrightarrow{a} is perpendicular to both overrightarrow{AB} and overrightarrow{AC}, its dot product with both vectors should be zero: overrightarrow{a} cdot overrightarrow{AB} = 2x = 0 Rightarrow x = 0 overrightarrow{a} cdot overrightarrow{AC} = 4y + 3z = 0 Furthermore, given |overrightarrow{a}| = 10, we apply the Pythagorean theorem: sqrt{x^2 + y^2 + z^2} = 10 Substitute x = 0 into the equation: sqrt{0 + y^2 + z^2} = 10 y^2 + z^2 = 100 Using the second condition 4y + 3z = 0, we can express y in terms of z: y = -frac{3}{4}z Substitute this into the previous equation to solve for z: left(-frac{3}{4}zright)^2 + z^2 = 100 frac{9}{16}z^2 + z^2 = 100 frac{25}{16}z^2 = 100 z^2 = frac{100 cdot 16}{25} z^2 = 64 So the solutions for z are z = 8 and z = -8. Conversely, the corresponding y values are y = -6 and y = 6 respectively. Therefore, we have two possible coordinates for vector overrightarrow{a} considering a positive or negative value for z: boxed{overrightarrow{a} = (0, -6, 8)} text{ or } boxed{overrightarrow{a} = (0, 6, -8)}

answer:To find the rate per square meter for paving the floor, we first need to calculate the total area of the floor that needs to be paved. The area of the floor can be calculated by multiplying the length by the width. Area = Length × Width Area = 5.5 m × 3.75 m Area = 20.625 square meters Now that we have the total area, we can calculate the rate per square meter by dividing the total cost by the total area. Rate per square meter = Total cost / Total area Rate per square meter = Rs. 24750 / 20.625 sq. m Rate per square meter = Rs. 1200 per sq. m Therefore, the rate per square meter for paving the floor is Rs. boxed{1200} .

answer:1. **Express N in terms of its digits**: Let N = 10t + u, where t is the tens digit and u is the units digit. This is a standard way to express a two-digit number in terms of its digits. 2. **Formulate the condition given in the problem**: We need to find when the sum of N and the number obtained by reversing its digits is a perfect square. If the digits are reversed, the number becomes 10u + t. Therefore, the sum of N and its reverse is: [ (10t + u) + (10u + t) = 11t + 11u = 11(t + u). ] 3. **Simplify the condition**: We have 11(t + u), and we want this to be a perfect square. Since 11 is a prime number, for 11(t + u) to be a perfect square, t + u must itself be a multiple of 11 (because the square root of 11(t + u) must be an integer, and 11 must be paired with another 11 in the factorization to contribute to a square). 4. **Constrain t + u**: Since t and u are digits (i.e., 0 leq t, u leq 9), the sum t + u can range from 0 to 18. The only multiple of 11 within this range is 11 itself. 5. **Determine possible digit pairs (t, u)**: We need t + u = 11. The possible pairs (t, u) satisfying this are (2, 9), (3, 8), (4, 7), (5, 6), (6, 5), (7, 4), (8, 3), (9, 2). 6. **Count the solutions**: There are 8 pairs of digits (t, u) that satisfy the condition. Thus, the number of two-digit integers N such that N plus its reverse is a perfect square is boxed{8}. This corresponds to choice textbf{(E)} 8.

answer:Since overrightarrow{MA} cdot overrightarrow{MB}=0, overrightarrow{AM} cdot overrightarrow{AB}=-overrightarrow{MA} cdot (overrightarrow{MB}-overrightarrow{MA})=overrightarrow{MA}^2. Let A(2cos{alpha}, sin{alpha}), then overrightarrow{MA}^2=(2cos{alpha}-1)^2+sin^2{alpha}=3cos^2{alpha}-4cos{alpha}+2=3(cos{alpha}-frac{2}{3})^2+frac{2}{3}. Thus, when cos{alpha}=frac{2}{3}, overrightarrow{MA}^2 reaches its minimum value of frac{2}{3}. Hence, the minimum value of overrightarrow{AM} cdot overrightarrow{AB} is boxed{frac{2}{3}}. Using overrightarrow{MA} cdot overrightarrow{MB}=0, we can derive that overrightarrow{AM} cdot overrightarrow{AB}=overrightarrow{MA}^2. By representing overrightarrow{MA}^2 with trigonometric functions containing alpha, we can find the minimum value of overrightarrow{AM} cdot overrightarrow{AB}.