💡Alright, so I have this problem where I'm given a vector a, which is (1, 0, -1), and I need to figure out which of the given vectors forms a 60-degree angle with vector a. The options are A: (-1,1,0), B: (1,-1,0), C: (0,-1,1), and D: (-1,0,1). First, I remember that the angle between two vectors can be found using the dot product formula. The formula is:cosθ = (a · b) / (|a| |b|)Where θ is the angle between vectors a and b, a · b is the dot product of a and b, and |a| and |b| are the magnitudes of vectors a and b, respectively.So, my goal is to compute the dot product of vector a with each of the given vectors, then divide that by the product of their magnitudes, and see which one gives me cosθ equal to cos60°, which is 0.5.Let me start by computing the magnitude of vector a. Vector a is (1, 0, -1), so its magnitude |a| is sqrt(1² + 0² + (-1)²) = sqrt(1 + 0 + 1) = sqrt(2). So, |a| = sqrt(2).Now, I'll go through each option one by one.**Option A: (-1, 1, 0)**First, compute the dot product a · b:a · b = (1)(-1) + (0)(1) + (-1)(0) = -1 + 0 + 0 = -1Next, compute the magnitude of vector b:|b| = sqrt((-1)² + 1² + 0²) = sqrt(1 + 1 + 0) = sqrt(2)Now, plug these into the formula:cosθ = (-1) / (sqrt(2) * sqrt(2)) = -1 / 2So, cosθ = -0.5, which means θ is 120 degrees, not 60. So, option A is not the answer.**Option B: (1, -1, 0)**Compute the dot product a · b:a · b = (1)(1) + (0)(-1) + (-1)(0) = 1 + 0 + 0 = 1Compute the magnitude of vector b:|b| = sqrt(1² + (-1)² + 0²) = sqrt(1 + 1 + 0) = sqrt(2)Now, plug into the formula:cosθ = 1 / (sqrt(2) * sqrt(2)) = 1 / 2So, cosθ = 0.5, which means θ is 60 degrees. This matches the condition given in the problem. So, option B seems to be the correct answer.But just to be thorough, I'll check the other options too.**Option C: (0, -1, 1)**Compute the dot product a · b:a · b = (1)(0) + (0)(-1) + (-1)(1) = 0 + 0 -1 = -1Compute the magnitude of vector b:|b| = sqrt(0² + (-1)² + 1²) = sqrt(0 + 1 + 1) = sqrt(2)Plug into the formula:cosθ = (-1) / (sqrt(2) * sqrt(2)) = -1 / 2Again, cosθ = -0.5, so θ is 120 degrees. Not the answer.**Option D: (-1, 0, 1)**Compute the dot product a · b:a · b = (1)(-1) + (0)(0) + (-1)(1) = -1 + 0 -1 = -2Compute the magnitude of vector b:|b| = sqrt((-1)² + 0² + 1²) = sqrt(1 + 0 + 1) = sqrt(2)Plug into the formula:cosθ = (-2) / (sqrt(2) * sqrt(2)) = -2 / 2 = -1So, cosθ = -1, which means θ is 180 degrees. Definitely not 60 degrees.So, after checking all options, only option B gives me a cosine of 0.5, which corresponds to a 60-degree angle. Therefore, the correct answer should be option B.

💡Okay, so I have this problem where I need to find the magnitude of the vector a plus twice the vector b. The vectors are given as a = (cos 5°, sin 5°) and b = (cos 65°, sin 65°). Hmm, okay. I remember that to find the magnitude of a vector, you use the formula |v| = sqrt(v_x² + v_y²). So, I guess I need to first find the components of the vector a + 2b and then calculate its magnitude.Let me break it down step by step. First, I need to compute 2b. Since b is (cos 65°, sin 65°), multiplying by 2 would just scale both components by 2. So, 2b would be (2cos 65°, 2sin 65°). Got that.Next, I need to add vector a to this. Vector a is (cos 5°, sin 5°). So, adding them component-wise, the x-component would be cos 5° + 2cos 65°, and the y-component would be sin 5° + 2sin 65°. So, the resulting vector is (cos 5° + 2cos 65°, sin 5° + 2sin 65°).Now, to find the magnitude of this resulting vector, I need to square both components, add them together, and then take the square root. So, let's write that out:|a + 2b| = sqrt[(cos 5° + 2cos 65°)² + (sin 5° + 2sin 65°)²]Hmm, that looks a bit complicated. Maybe I can simplify this expression somehow. I remember that when you have expressions like (A + B)², you can expand them into A² + 2AB + B². Maybe I can apply that here.Let me expand the x-component squared:(cos 5° + 2cos 65°)² = cos² 5° + 4cos 5°cos 65° + 4cos² 65°Similarly, the y-component squared:(sin 5° + 2sin 65°)² = sin² 5° + 4sin 5°sin 65° + 4sin² 65°Now, if I add these two together, I get:cos² 5° + sin² 5° + 4cos 5°cos 65° + 4sin 5°sin 65° + 4cos² 65° + 4sin² 65°I remember that cos² θ + sin² θ = 1, so cos² 5° + sin² 5° = 1, and similarly, 4cos² 65° + 4sin² 65° = 4(cos² 65° + sin² 65°) = 4*1 = 4.So, substituting that back in, the expression becomes:1 + 4 + 4cos 5°cos 65° + 4sin 5°sin 65°Which simplifies to:5 + 4(cos 5°cos 65° + sin 5°sin 65°)Wait a second, cos 5°cos 65° + sin 5°sin 65° looks familiar. Isn't that the formula for cos(A - B)? Yes, because cos(A - B) = cos A cos B + sin A sin B. So, in this case, A is 5° and B is 65°, so it becomes cos(5° - 65°) = cos(-60°). But cosine is an even function, so cos(-60°) = cos(60°) = 0.5.So, substituting that back in, we have:5 + 4*(0.5) = 5 + 2 = 7Therefore, the magnitude |a + 2b| is sqrt(7).Wait, let me double-check my steps to make sure I didn't make a mistake. First, I expanded the squares correctly, then used the Pythagorean identity to simplify the cos² and sin² terms. Then, I recognized the cosine of difference identity correctly and evaluated cos(-60°) as 0.5. That seems right. So, yes, the magnitude should be sqrt(7).

💡First, I need to determine the total number of ways to choose any four dots from the 25 dots in the 5x5 grid. This can be calculated using the combination formula:[text{Total combinations} = binom{25}{4} = frac{25!}{4!(25-4)!} = 12650]Next, I'll count the number of favorable outcomes where the four dots form a horizontal line. There are 5 horizontal lines in the grid, and each line has exactly one set of four collinear dots:[text{Horizontal lines} = 5]Similarly, I'll count the number of favorable outcomes where the four dots form a vertical line. There are also 5 vertical lines in the grid, with each line having one set of four collinear dots:[text{Vertical lines} = 5]Adding these together gives the total number of favorable outcomes:[text{Total favorable outcomes} = 5 + 5 = 10]Finally, I'll calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:[text{Probability} = frac{10}{12650} = frac{1}{1265}]

💡To find the number, let's denote it as ( x ).According to the problem, subtracting ( x ) from 150 gives the same result as adding ( x ) to 68. This can be written as the equation:[150 - x = x + 68]To solve for ( x ), we'll first move all terms involving ( x ) to one side and constants to the other side:[150 - 68 = x + x][82 = 2x]Next, we'll divide both sides by 2 to isolate ( x ):[x = frac{82}{2} = 41]Therefore, the number is ( 41 ).

💡First, I need to determine how many oranges Michaela needs to eat until she gets full. Let's denote this number as ( M ).Cassandra needs twice as many oranges as Michaela to get full, so Cassandra needs ( 2M ) oranges.Together, Michaela and Cassandra picked 90 oranges. After eating until they were full, 30 oranges remained. This means they consumed a total of ( 90 - 30 = 60 ) oranges.The total number of oranges eaten by both Michaela and Cassandra is the sum of the oranges each of them ate:[M + 2M = 3M]Since they ate 60 oranges in total, we can set up the equation:[3M = 60]To find the value of ( M ), we divide both sides of the equation by 3:[M = frac{60}{3} = 20]Therefore, Michaela needs to eat (boxed{20}) oranges to get full.

💡First, I need to determine the daily production rate for each type of weaver based on the information provided.For the mat-weavers:- 4 mat-weavers can weave 4 mats in 4 days.- This means 1 mat-weaver can weave 1 mat in 4 days.- Therefore, the daily production rate per mat-weaver is 1 mat divided by 4 days, which equals 0.25 mats per day.- If 5 mat-weavers work for 8 days, they will weave 5 mat-weavers multiplied by 0.25 mats per day per mat-weaver, then multiplied by 8 days, resulting in 10 mats.Next, for the carpet-weavers:- 3 carpet-weavers can weave 2 carpets in 3 days.- This means 1 carpet-weaver can weave 2/3 of a carpet in 3 days.- The daily production rate per carpet-weaver is (2/3) carpets divided by 3 days, which equals 2/9 carpets per day.- If 6 carpet-weavers work for 8 days, they will weave 6 carpet-weavers multiplied by (2/9) carpets per day per carpet-weaver, then multiplied by 8 days, resulting in approximately 10.67 carpets. Since we can't have a fraction of a carpet, we'll round down to 10 carpets.For the rug-weavers:- 2 rug-weavers can weave 3 rugs in 5 days.- This means 1 rug-weaver can weave 3/2 rugs in 5 days.- The daily production rate per rug-weaver is (3/2) rugs divided by 5 days, which equals 3/10 rugs per day.- If 4 rug-weavers work for 8 days, they will weave 4 rug-weavers multiplied by (3/10) rugs per day per rug-weaver, then multiplied by 8 days, resulting in approximately 9.6 rugs. Rounding down, they will weave 9 rugs.Finally, for the curtain-weavers:- 1 curtain-weaver can create 1 curtain in 6 days.- The daily production rate per curtain-weaver is 1 curtain divided by 6 days, which is approximately 0.1667 curtains per day.- If 2 curtain-weavers work for 8 days, they will create 2 curtain-weavers multiplied by 0.1667 curtains per day per curtain-weaver, then multiplied by 8 days, resulting in approximately 2.667 curtains. Rounding down, they will create 2 curtains.In summary, over an 8-day period:- The mat-weavers will weave 10 mats.- The carpet-weavers will weave 10 carpets.- The rug-weavers will weave 9 rugs.- The curtain-weavers will create 2 curtains.