💡Okay, so I have this problem here about an arithmetic sequence. It says that the sum of the first n terms is Sₙ, and it gives me the equation 2S₃ - 3S₂ = 12. I need to find the common difference of the sequence. Hmm, arithmetic sequences, right? Let me recall what I know about them.First, an arithmetic sequence is a sequence where each term after the first is obtained by adding a constant difference to the preceding term. That constant is called the common difference, usually denoted by d. So, if the first term is a₁, then the second term is a₁ + d, the third term is a₁ + 2d, and so on.Now, the sum of the first n terms of an arithmetic sequence, Sₙ, can be calculated using the formula: Sₙ = n/2 [2a₁ + (n - 1)d]. Alternatively, it's also written as Sₙ = n(a₁ + aₙ)/2, where aₙ is the nth term. I think both formulas are equivalent because aₙ = a₁ + (n - 1)d.Alright, so the problem gives me 2S₃ - 3S₂ = 12. I need to find d. Let me write down what S₂ and S₃ are using the sum formula.For S₂, which is the sum of the first 2 terms:S₂ = 2/2 [2a₁ + (2 - 1)d] = 1 [2a₁ + d] = 2a₁ + d.Similarly, for S₃, the sum of the first 3 terms:S₃ = 3/2 [2a₁ + (3 - 1)d] = (3/2)(2a₁ + 2d) = 3(a₁ + d).Wait, let me double-check that. If I use the other formula, Sₙ = n(a₁ + aₙ)/2, then for S₃, a₃ = a₁ + 2d, so S₃ = 3(a₁ + a₃)/2 = 3(a₁ + a₁ + 2d)/2 = 3(2a₁ + 2d)/2 = 3(a₁ + d). Yep, that's the same as before.So, S₂ = 2a₁ + d and S₃ = 3a₁ + 3d. Got that.Now, plug these into the equation 2S₃ - 3S₂ = 12.Let me compute 2S₃ first: 2*(3a₁ + 3d) = 6a₁ + 6d.Then, compute 3S₂: 3*(2a₁ + d) = 6a₁ + 3d.Subtracting these: 2S₃ - 3S₂ = (6a₁ + 6d) - (6a₁ + 3d) = 6a₁ + 6d - 6a₁ - 3d = 3d.So, 3d = 12. Therefore, d = 12 / 3 = 4.Wait, that seems straightforward. So, the common difference d is 4. Let me just verify my steps to make sure I didn't make a mistake.1. Calculated S₂ and S₃ correctly using the sum formula.2. Plugged them into the given equation.3. Expanded and simplified the equation correctly.4. Solved for d and got 4.I think that's all correct. I don't see any errors in my calculations. So, the common difference is 4.**Final Answer**The common difference of the sequence is boxed{4}.

💡First, I'll denote the weights of A, B, C, and D as ( W_A, W_B, W_C, ) and ( W_D ) respectively. Let ( W_E ) be the weight of E.Given that the average weight of A, B, and C is 84 kg, I can write the equation:[frac{W_A + W_B + W_C}{3} = 84]Multiplying both sides by 3, I get:[W_A + W_B + W_C = 252 quad text{(1)}]When D joins the group, the average weight becomes 80 kg. So, for A, B, C, and D together:[frac{W_A + W_B + W_C + W_D}{4} = 80]Multiplying both sides by 4, I get:[W_A + W_B + W_C + W_D = 320 quad text{(2)}]Subtracting equation (1) from equation (2), I find the weight of D:[W_D = 320 - 252 = 68 quad text{(3)}]When E replaces A, the average weight of B, C, D, and E becomes 79 kg. So:[frac{W_B + W_C + W_D + W_E}{4} = 79]Multiplying both sides by 4, I get:[W_B + W_C + W_D + W_E = 316 quad text{(4)}]Given ( W_A = 80 ) kg, I substitute ( W_A ) into equation (1) to find ( W_B + W_C ):[80 + W_B + W_C = 252][W_B + W_C = 172 quad text{(5)}]Substituting ( W_B + W_C ) and ( W_D ) into equation (4):[172 + 68 + W_E = 316][240 + W_E = 316][W_E = 76 quad text{(6)}]Finally, the difference in weight between E and D is:[W_E - W_D = 76 - 68 = 8]

💡Alright, so I have this geometry problem here about skew lines and their relationship with a plane. Let me try to unpack it step by step. The question is: If lines a and b are skew, and line a is parallel to plane α, then what is the positional relationship between line b and plane α? The options are parallel, intersecting, b is within α, or any of these possibilities.First, I need to recall what skew lines are. Skew lines are lines that do not intersect and are not parallel; they exist in different planes. So, lines a and b are not in the same plane, and they don't intersect. Now, line a is parallel to plane α. What does that mean? If a line is parallel to a plane, it means that the line doesn't intersect the plane and is not lying on the plane. So, line a is outside plane α and doesn't touch it.Now, I need to figure out what this implies about line b. Since a and b are skew, they are not parallel, and they don't intersect. But line a is parallel to plane α. So, line b could be in any position relative to plane α, right? It could be parallel, intersecting, or even lying on the plane.Wait, but line a is parallel to plane α, and a and b are skew. If line b were lying on plane α, would that affect the skewness of a and b? Let me think. If b is on plane α, and a is parallel to α, then a and b would still be skew because they don't intersect and aren't parallel. So, that's possible.Similarly, if line b is parallel to plane α, then it doesn't intersect α and isn't on it. That would also keep a and b skew because they don't intersect and aren't parallel. Alternatively, if line b intersects plane α, then it would cross the plane at some point, but since a is parallel to α, a and b would still be skew because they don't intersect each other.So, it seems like line b can have multiple relationships with plane α: it can be parallel, intersecting, or lying on the plane. Therefore, the correct answer should include all these possibilities.Let me double-check. If a is parallel to α, and a and b are skew, then b can't be parallel to a because skew lines aren't parallel. But b can be parallel to α, intersecting α, or lying on α. So, yeah, all three options are possible. Therefore, the answer should be D: Parallel, intersecting, or b is within α.I think that makes sense. I don't see any contradictions in this reasoning. So, I'm pretty confident that D is the correct choice.

💡Alright, so I've got this problem about elderly apartments and probability. It's divided into two parts, and each part has a couple of sub-questions. Let me try to break it down step by step.Starting with part (1):We have 120 elderly residents in total, distributed across single, double, and triple rooms. The numbers are 36 in single rooms, 60 in double rooms, and 24 in triple rooms. The first task is to use stratified sampling to select 10 people based on their room types, and then from those 10, randomly select 4 people. We need to find the probability distribution and the expected value of the number of single room residents among these 4 people, denoted as ξ.Okay, so stratified sampling means we'll select a proportionate number of residents from each room type. Let's figure out how many from each room type will be in the sample of 10.First, calculate the proportion of each room type:- Single rooms: 36/120 = 0.3- Double rooms: 60/120 = 0.5- Triple rooms: 24/120 = 0.2So, for 10 people:- Single rooms: 0.3 * 10 = 3- Double rooms: 0.5 * 10 = 5- Triple rooms: 0.2 * 10 = 2So, in the sample of 10, there are 3 single room residents, 5 double room residents, and 2 triple room residents.Now, from these 10, we select 4 people. We need to find the probability distribution of ξ, which is the number of single room residents in these 4.This sounds like a hypergeometric distribution problem because we're sampling without replacement from a finite population with two groups: single room residents and others.The hypergeometric distribution formula is:P(ξ = k) = [C(K, k) * C(N-K, n-k)] / C(N, n)Where:- N = total population size = 10- K = number of success states in the population = 3 (single room residents)- n = number of draws = 4- k = number of observed successes = 0,1,2,3So, let's compute P(ξ = k) for k = 0,1,2,3.First, compute the combinations:C(3,0) * C(7,4) / C(10,4)C(3,1) * C(7,3) / C(10,4)C(3,2) * C(7,2) / C(10,4)C(3,3) * C(7,1) / C(10,4)Calculating each:C(10,4) = 210For k=0:C(3,0)=1, C(7,4)=35So, 1*35 / 210 = 35/210 = 1/6 ≈ 0.1667For k=1:C(3,1)=3, C(7,3)=35So, 3*35 / 210 = 105/210 = 1/2 = 0.5For k=2:C(3,2)=3, C(7,2)=21So, 3*21 / 210 = 63/210 = 3/10 = 0.3For k=3:C(3,3)=1, C(7,1)=7So, 1*7 / 210 = 7/210 = 1/30 ≈ 0.0333Let me check if these probabilities add up to 1:1/6 + 1/2 + 3/10 + 1/30 ≈ 0.1667 + 0.5 + 0.3 + 0.0333 ≈ 1Good, that seems correct.So, the probability distribution is:ξ | 0 | 1 | 2 | 3---|---|---|---|---P(ξ) | 1/6 | 1/2 | 3/10 | 1/30Now, the expected value E[ξ] can be calculated as:E[ξ] = Σ [k * P(ξ = k)] for k=0 to 3So,E[ξ] = 0*(1/6) + 1*(1/2) + 2*(3/10) + 3*(1/30)Calculating each term:0 + 1/2 + 6/10 + 3/30Simplify:1/2 = 0.56/10 = 0.63/30 = 0.1Adding them up: 0.5 + 0.6 + 0.1 = 1.2So, E[ξ] = 6/5 = 1.2Okay, that seems reasonable.Moving on to part (2):We're told to consider double and triple rooms as multiple occupancy rooms. So, single rooms are 36, and multiple occupancy rooms are 60 + 24 = 84.During opinion collection, we need to group 2 single room residents and m multiple occupancy residents together, where m > 2 and m is a positive integer. Then, from this group, we randomly select 2 people. If both have the same room type, the group is labeled I; otherwise, it's labeled II. We need to find the probability p that a group is labeled II.First, let's understand the group composition:- 2 single room residents- m multiple occupancy residentsTotal group size = 2 + mWhen selecting 2 people, the total number of ways is C(2 + m, 2).The number of ways to select 2 people with the same room type:- Both single: C(2, 2) = 1- Both multiple: C(m, 2) = m(m - 1)/2So, total same room type selections: 1 + m(m - 1)/2Therefore, the probability of selecting two people with the same room type is:P(same) = [1 + m(m - 1)/2] / C(2 + m, 2)But the group is labeled II if the two people have different room types, so:p = 1 - P(same)Let me compute P(same):C(2 + m, 2) = (2 + m)(1 + m)/2So,P(same) = [1 + m(m - 1)/2] / [(2 + m)(1 + m)/2]Simplify numerator:1 + [m(m - 1)/2] = [2 + m(m - 1)] / 2So,P(same) = [2 + m(m - 1)] / 2 divided by [(2 + m)(1 + m)/2] = [2 + m(m - 1)] / [(2 + m)(1 + m)]Simplify numerator:2 + m^2 - m = m^2 - m + 2Denominator:(2 + m)(1 + m) = m^2 + 3m + 2So,P(same) = (m^2 - m + 2)/(m^2 + 3m + 2)Therefore,p = 1 - (m^2 - m + 2)/(m^2 + 3m + 2) = [ (m^2 + 3m + 2) - (m^2 - m + 2) ] / (m^2 + 3m + 2)Simplify numerator:m^2 + 3m + 2 - m^2 + m - 2 = 4mSo,p = 4m / (m^2 + 3m + 2)That's the expression for p in terms of m.Now, part (ii):We have 5 groups, and g(p) is the probability that exactly 3 groups are labeled II. We need to find the maximum value of g(p) and the corresponding m.First, g(p) is a binomial probability:g(p) = C(5,3) * p^3 * (1 - p)^2Simplify:g(p) = 10 * p^3 * (1 - p)^2We need to maximize this function with respect to p, but p itself is a function of m: p = 4m / (m^2 + 3m + 2)So, we need to find the value of m that maximizes g(p(m)).Alternatively, since m is an integer greater than 2, we can compute g(p) for m = 3,4,5,... and find which gives the maximum.But perhaps we can first find the value of p that maximizes g(p), and then find m that gives that p.Let's treat g(p) as a function of p:g(p) = 10 p^3 (1 - p)^2To find its maximum, take derivative with respect to p and set to zero.g'(p) = 10 [3p^2 (1 - p)^2 + p^3 * 2(1 - p)(-1)]Simplify:= 10 [3p^2 (1 - p)^2 - 2p^3 (1 - p)]Factor out p^2 (1 - p):= 10 p^2 (1 - p) [3(1 - p) - 2p]= 10 p^2 (1 - p) [3 - 3p - 2p]= 10 p^2 (1 - p) [3 - 5p]Set derivative to zero:10 p^2 (1 - p) (3 - 5p) = 0Solutions are p = 0, p = 1, and 3 - 5p = 0 => p = 3/5Since p = 0 and p = 1 give g(p) = 0, the maximum must be at p = 3/5.So, the maximum of g(p) occurs at p = 3/5.Now, we need to find m such that p = 4m / (m^2 + 3m + 2) = 3/5Solve for m:4m / (m^2 + 3m + 2) = 3/5Cross-multiplying:20m = 3(m^2 + 3m + 2)20m = 3m^2 + 9m + 6Bring all terms to one side:3m^2 + 9m + 6 - 20m = 03m^2 - 11m + 6 = 0Solve quadratic equation:m = [11 ± sqrt(121 - 72)] / 6 = [11 ± sqrt(49)] / 6 = [11 ± 7] / 6So,m = (11 + 7)/6 = 18/6 = 3m = (11 - 7)/6 = 4/6 = 2/3But m must be an integer greater than 2, so m = 3 is the solution.Therefore, when m = 3, p = 3/5, and g(p) is maximized.Compute g(3/5):g(3/5) = 10*(3/5)^3*(2/5)^2 = 10*(27/125)*(4/25) = 10*(108/3125) = 1080/3125 = 216/625Simplify: 216 ÷ 625 = 0.3456So, the maximum value of g(p) is 216/625 when m = 3.Let me double-check the calculations:For m=3,p = 4*3 / (9 + 9 + 2) = 12 / 20 = 3/5Yes, that's correct.Then, g(p) = 10*(3/5)^3*(2/5)^2 = 10*(27/125)*(4/25) = 10*(108/3125) = 1080/3125 = 216/625Yes, that's correct.So, the maximum value is 216/625 when m=3.**Final Answer**(1) The probability distribution of xi is:| xi | 0 | 1 | 2 | 3 ||-------|---|---|---|---|| P | frac{1}{6} | frac{1}{2} | frac{3}{10} | frac{1}{30} |The mathematical expectation is boxed{dfrac{6}{5}}.(2)(i) The probability p is boxed{dfrac{4m}{m^2 + 3m + 2}}.(2)(ii) The maximum value of g(p) is boxed{dfrac{216}{625}} when m = boxed{3}.

💡Okay, so I need to find the monotonic increasing interval for the function ( f(x) = log_3 (x^2 - 2x - 3) ). Hmm, let me think about how to approach this step by step.First, I remember that for a logarithmic function like ( log_b (g(x)) ), the function is defined only when the argument ( g(x) ) is positive. So, I need to find the domain of ( f(x) ) by solving the inequality ( x^2 - 2x - 3 > 0 ).Let me solve the quadratic inequality ( x^2 - 2x - 3 > 0 ). To do this, I'll first find the roots of the quadratic equation ( x^2 - 2x - 3 = 0 ). Using the quadratic formula:[x = frac{2 pm sqrt{(-2)^2 - 4 cdot 1 cdot (-3)}}{2 cdot 1} = frac{2 pm sqrt{4 + 12}}{2} = frac{2 pm sqrt{16}}{2} = frac{2 pm 4}{2}]So, the roots are ( x = frac{2 + 4}{2} = 3 ) and ( x = frac{2 - 4}{2} = -1 ).Now, to determine where the quadratic expression ( x^2 - 2x - 3 ) is positive, I can analyze the intervals determined by the roots: ( (-infty, -1) ), ( (-1, 3) ), and ( (3, infty) ).Let me pick test points in each interval:1. For ( x in (-infty, -1) ), let's choose ( x = -2 ): [ (-2)^2 - 2(-2) - 3 = 4 + 4 - 3 = 5 > 0 ] So, the expression is positive in this interval.2. For ( x in (-1, 3) ), let's choose ( x = 0 ): [ 0^2 - 2(0) - 3 = -3 < 0 ] The expression is negative here.3. For ( x in (3, infty) ), let's choose ( x = 4 ): [ 4^2 - 2(4) - 3 = 16 - 8 - 3 = 5 > 0 ] The expression is positive in this interval.Therefore, the domain of ( f(x) ) is ( (-infty, -1) cup (3, infty) ).Next, I need to determine where the function ( f(x) = log_3 (x^2 - 2x - 3) ) is increasing. Since the logarithm function ( log_3 (t) ) is an increasing function for ( t > 0 ), the behavior of ( f(x) ) depends on the behavior of the quadratic function ( g(x) = x^2 - 2x - 3 ).To find where ( f(x) ) is increasing, I should look at where ( g(x) ) is increasing because the composition of two increasing functions is increasing. Conversely, if ( g(x) ) is decreasing, then ( f(x) ) would be decreasing as well.Let me find the derivative of ( g(x) ) to determine its intervals of increase and decrease.[g'(x) = 2x - 2]Setting ( g'(x) = 0 ) to find critical points:[2x - 2 = 0 implies x = 1]So, the critical point is at ( x = 1 ). Now, I'll analyze the sign of ( g'(x) ) in the intervals determined by ( x = 1 ):1. For ( x < 1 ), say ( x = 0 ): [ g'(0) = 2(0) - 2 = -2 < 0 ] So, ( g(x) ) is decreasing on ( (-infty, 1) ).2. For ( x > 1 ), say ( x = 2 ): [ g'(2) = 2(2) - 2 = 2 > 0 ] So, ( g(x) ) is increasing on ( (1, infty) ).But wait, the domain of ( f(x) ) is ( (-infty, -1) cup (3, infty) ), so I need to consider the behavior of ( g(x) ) only within these intervals.In the interval ( (-infty, -1) ), since ( -infty < -1 < 1 ), ( g(x) ) is decreasing here.In the interval ( (3, infty) ), since ( 3 > 1 ), ( g(x) ) is increasing here.Therefore, since ( f(x) = log_3 (g(x)) ) is a composition of the increasing logarithm function and ( g(x) ), the function ( f(x) ) will be increasing where ( g(x) ) is increasing, and decreasing where ( g(x) ) is decreasing.So, in the interval ( (-infty, -1) ), ( g(x) ) is decreasing, so ( f(x) ) is decreasing.In the interval ( (3, infty) ), ( g(x) ) is increasing, so ( f(x) ) is increasing.Therefore, the monotonic increasing interval for ( f(x) ) is ( (3, infty) ).I should double-check my work to make sure I didn't make any mistakes.1. Domain: ( x^2 - 2x - 3 > 0 ) leads to ( x in (-infty, -1) cup (3, infty) ). That seems correct.2. Derivative of ( g(x) ): ( g'(x) = 2x - 2 ). Critical point at ( x = 1 ). So, ( g(x) ) decreasing on ( (-infty, 1) ) and increasing on ( (1, infty) ). That makes sense.3. Within the domain of ( f(x) ), ( g(x) ) is decreasing on ( (-infty, -1) ) and increasing on ( (3, infty) ). Thus, ( f(x) ) is decreasing on ( (-infty, -1) ) and increasing on ( (3, infty) ).Yes, everything checks out. So, the function ( f(x) = log_3 (x^2 - 2x - 3) ) is increasing on ( (3, infty) ).**Final Answer**The monotonic increasing interval for the function is boxed{(3, +infty)}.

💡Okay, so I have this cube, right? It's labeled ABCD-A₁B₁C₁D₁, and each edge has a length of 2. I need to find the dot product of two vectors: vector AA₁ and vector BC₁. Hmm, okay, let me think about how to approach this.First, I remember that in a cube, all edges are equal, and all angles are right angles. So, AA₁ is one of the vertical edges going from the base to the top face, and BC₁ is a diagonal edge from B to C₁. I need to find their dot product.I think the best way to tackle this is by using coordinates. If I assign coordinates to each vertex of the cube, I can express the vectors AA₁ and BC₁ in terms of their components and then compute the dot product.Let me set up a coordinate system. Let's place point A at the origin (0, 0, 0). Since the cube has edge length 2, the coordinates of the other points can be determined accordingly.So, point A is (0, 0, 0). Moving along the x-axis, point B would be (2, 0, 0). Then, point C is (2, 2, 0), and point D is (0, 2, 0). Now, the top face points: A₁ is directly above A, so that's (0, 0, 2). Similarly, B₁ is (2, 0, 2), C₁ is (2, 2, 2), and D₁ is (0, 2, 2).Alright, now let me write down the coordinates:- A: (0, 0, 0)- B: (2, 0, 0)- C: (2, 2, 0)- D: (0, 2, 0)- A₁: (0, 0, 2)- B₁: (2, 0, 2)- C₁: (2, 2, 2)- D₁: (0, 2, 2)Great, now I can find the vectors AA₁ and BC₁.Vector AA₁ goes from A to A₁. So, subtracting the coordinates of A from A₁:AA₁ = A₁ - A = (0 - 0, 0 - 0, 2 - 0) = (0, 0, 2)Similarly, vector BC₁ goes from B to C₁. So, subtracting the coordinates of B from C₁:BC₁ = C₁ - B = (2 - 2, 2 - 0, 2 - 0) = (0, 2, 2)Wait, hold on. Let me double-check that. From B to C₁, in the cube, that's moving from (2, 0, 0) to (2, 2, 2). So, the change in x is 0, change in y is 2, and change in z is 2. So yes, BC₁ is (0, 2, 2). Okay, that seems right.Now, to find the dot product of AA₁ and BC₁, I can use the formula for the dot product of two vectors:If vector u = (u₁, u₂, u₃) and vector v = (v₁, v₂, v₃), thenu · v = u₁v₁ + u₂v₂ + u₃v₃So, applying this to AA₁ and BC₁:AA₁ = (0, 0, 2)BC₁ = (0, 2, 2)Dot product = (0)(0) + (0)(2) + (2)(2) = 0 + 0 + 4 = 4Wait, that seems straightforward. So, the dot product is 4. Looking back at the options, D is 4. So, is that the answer?But let me think again, just to make sure I didn't make any mistakes. Maybe I assigned the coordinates incorrectly or miscalculated the vectors.Let me verify the coordinates:- Starting from A (0,0,0), moving along x to B (2,0,0), then y to C (2,2,0), then back to D (0,2,0). Then up to A₁ (0,0,2), and similarly for the others. That seems correct.Vectors:- AA₁ is from A (0,0,0) to A₁ (0,0,2): (0,0,2). Correct.- BC₁ is from B (2,0,0) to C₁ (2,2,2): subtracting, (0,2,2). Correct.Dot product calculation:0*0 + 0*2 + 2*2 = 0 + 0 + 4 = 4. Yep, that's right.Alternatively, maybe I can think about this geometrically. The dot product is also equal to the product of the magnitudes of the vectors and the cosine of the angle between them.So, |AA₁| is the length of AA₁, which is 2, since it's an edge of the cube.|BC₁| is the length of BC₁. Let's compute that. From B (2,0,0) to C₁ (2,2,2). The differences are 0 in x, 2 in y, and 2 in z. So, the length is sqrt(0² + 2² + 2²) = sqrt(0 + 4 + 4) = sqrt(8) = 2√2.So, |AA₁| = 2, |BC₁| = 2√2.Now, the angle between AA₁ and BC₁. Hmm, in the cube, AA₁ is vertical, and BC₁ is a diagonal going from B to C₁, which is on the top face.Let me visualize this. AA₁ is along the z-axis, and BC₁ is going from (2,0,0) to (2,2,2). So, BC₁ has components in the y and z directions.The angle between AA₁ and BC₁ can be found using the dot product formula:AA₁ · BC₁ = |AA₁||BC₁|cosθWe already found the dot product is 4, so:4 = (2)(2√2)cosθSimplify:4 = 4√2 cosθDivide both sides by 4√2:cosθ = 4 / (4√2) = 1/√2So, θ = 45 degrees.Wait, that's interesting. So, the angle between AA₁ and BC₁ is 45 degrees. That makes sense because BC₁ is a face diagonal on the top face, and AA₁ is vertical, so the projection of BC₁ onto the vertical direction would be equal to its y-component, which is 2, but wait, no, actually, the projection would involve the angle.But regardless, since we've already computed the dot product as 4, and the magnitude method also gives us 4, that seems consistent.Alternatively, maybe I can think about the vectors in terms of unit vectors.Let me define the standard basis vectors:i = (1,0,0), j = (0,1,0), k = (0,0,1)Then, vector AA₁ is (0,0,2), which is 2k.Vector BC₁ is (0,2,2), which is 2j + 2k.So, their dot product is (2k) · (2j + 2k) = 2k · 2j + 2k · 2kDot product is distributive, so:= (2*2)(k · j) + (2*2)(k · k)But k · j is 0 because they are perpendicular, and k · k is 1 because it's the same vector.So, this becomes 0 + 4*1 = 4.Yep, same result.So, all methods lead to the dot product being 4. Therefore, the answer should be D: 4.I think I've considered this problem from multiple angles: coordinate-based calculation, geometric interpretation using magnitudes and angles, and decomposition into basis vectors. All approaches confirm that the dot product is 4. So, I feel confident that D is the correct answer.**Final Answer**boxed{D}